Henrywood and Agarwal, Equation (12)

Percentage Accurate: 66.9% → 79.1%

Time: 30.5s

Alternatives: 23

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 66.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \end{array} \]

FPCore

(FPCore (d h l M D)
 :precision binary64
 (*
  (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0)))
  (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))

C

double code(double d, double h, double l, double M, double D) {
	return (pow((d / h), (1.0 / 2.0)) * pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Fortran

real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = (((d / h) ** (1.0d0 / 2.0d0)) * ((d / l) ** (1.0d0 / 2.0d0))) * (1.0d0 - (((1.0d0 / 2.0d0) * (((m * d_1) / (2.0d0 * d)) ** 2.0d0)) * (h / l)))
end function

Java

public static double code(double d, double h, double l, double M, double D) {
	return (Math.pow((d / h), (1.0 / 2.0)) * Math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * Math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)));
}

Python

def code(d, h, l, M, D):
	return (math.pow((d / h), (1.0 / 2.0)) * math.pow((d / l), (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * math.pow(((M * D) / (2.0 * d)), 2.0)) * (h / l)))

Julia

function code(d, h, l, M, D)
	return Float64(Float64((Float64(d / h) ^ Float64(1.0 / 2.0)) * (Float64(d / l) ^ Float64(1.0 / 2.0))) * Float64(1.0 - Float64(Float64(Float64(1.0 / 2.0) * (Float64(Float64(M * D) / Float64(2.0 * d)) ^ 2.0)) * Float64(h / l))))
end

MATLAB

function tmp = code(d, h, l, M, D)
	tmp = (((d / h) ^ (1.0 / 2.0)) * ((d / l) ^ (1.0 / 2.0))) * (1.0 - (((1.0 / 2.0) * (((M * D) / (2.0 * d)) ^ 2.0)) * (h / l)));
end

Wolfram

code[d_, h_, l_, M_, D_] := N[(N[(N[Power[N[(d / h), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], N[(1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(N[(1.0 / 2.0), $MachinePrecision] * N[Power[N[(N[(M * D), $MachinePrecision] / N[(2.0 * d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;{\left({\left(\frac{d}{h}\right)}^{0.25}\right)}^{2} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l -5e-310)
   (*
    (pow (pow (/ d h) 0.25) 2.0)
    (*
     (/ (sqrt (- d)) (sqrt (- l)))
     (- 1.0 (* 0.5 (/ (* h (pow (* (* 0.5 M) (/ D d)) 2.0)) l)))))
   (*
    (/ (sqrt d) (sqrt l))
    (*
     (/ (sqrt d) (sqrt h))
     (+ 1.0 (* (/ h l) (* (pow (* (/ D d) (/ M 2.0)) 2.0) -0.5)))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5e-310) {
		tmp = pow(pow((d / h), 0.25), 2.0) * ((sqrt(-d) / sqrt(-l)) * (1.0 - (0.5 * ((h * pow(((0.5 * M) * (D / d)), 2.0)) / l))));
	} else {
		tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * (1.0 + ((h / l) * (pow(((D / d) * (M / 2.0)), 2.0) * -0.5))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= (-5d-310)) then
        tmp = (((d / h) ** 0.25d0) ** 2.0d0) * ((sqrt(-d) / sqrt(-l)) * (1.0d0 - (0.5d0 * ((h * (((0.5d0 * m) * (d_1 / d)) ** 2.0d0)) / l))))
    else
        tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * (1.0d0 + ((h / l) * ((((d_1 / d) * (m / 2.0d0)) ** 2.0d0) * (-0.5d0)))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= -5e-310) {
		tmp = Math.pow(Math.pow((d / h), 0.25), 2.0) * ((Math.sqrt(-d) / Math.sqrt(-l)) * (1.0 - (0.5 * ((h * Math.pow(((0.5 * M) * (D / d)), 2.0)) / l))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * ((Math.sqrt(d) / Math.sqrt(h)) * (1.0 + ((h / l) * (Math.pow(((D / d) * (M / 2.0)), 2.0) * -0.5))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= -5e-310:
		tmp = math.pow(math.pow((d / h), 0.25), 2.0) * ((math.sqrt(-d) / math.sqrt(-l)) * (1.0 - (0.5 * ((h * math.pow(((0.5 * M) * (D / d)), 2.0)) / l))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(l)) * ((math.sqrt(d) / math.sqrt(h)) * (1.0 + ((h / l) * (math.pow(((D / d) * (M / 2.0)), 2.0) * -0.5))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= -5e-310)
		tmp = Float64(((Float64(d / h) ^ 0.25) ^ 2.0) * Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-l))) * Float64(1.0 - Float64(0.5 * Float64(Float64(h * (Float64(Float64(0.5 * M) * Float64(D / d)) ^ 2.0)) / l)))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(Float64(sqrt(d) / sqrt(h)) * Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0) * -0.5)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= -5e-310)
		tmp = (((d / h) ^ 0.25) ^ 2.0) * ((sqrt(-d) / sqrt(-l)) * (1.0 - (0.5 * ((h * (((0.5 * M) * (D / d)) ^ 2.0)) / l))));
	else
		tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * (1.0 + ((h / l) * ((((D / d) * (M / 2.0)) ^ 2.0) * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, -5e-310], N[(N[Power[N[Power[N[(d / h), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-l)], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h * N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.999999999999985e-310

   1. Initial program 70.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 69.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. pow1/2 69.2%

         \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. metadata-eval 69.2%

         \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      3. sqr-pow 69.2%

         \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      4. pow2 69.2%

         \[\leadsto \color{blue}{{\left({\left(\frac{d}{h}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      5. metadata-eval 69.2%

         \[\leadsto {\left({\left(\frac{d}{h}\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)}\right)}^{2} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      6. metadata-eval 69.2%

         \[\leadsto {\left({\left(\frac{d}{h}\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{{\left({\left(\frac{d}{h}\right)}^{0.25}\right)}^{2}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ 71.1%

         \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.25}\right)}^{2} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right)\right) \]

      2. add-sqr-sqrt 44.1%

         \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.25}\right)}^{2} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)}}^{2} \cdot h}{\ell}\right)\right) \]

      3. add-sqr-sqrt 71.1%

         \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.25}\right)}^{2} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot h}{\ell}\right)\right) \]

      4. div-inv 71.1%

         \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.25}\right)}^{2} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}\right)\right) \]

      5. metadata-eval 71.1%

         \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.25}\right)}^{2} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}\right)\right) \]

   7. Applied egg-rr 71.1%

      \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.25}\right)}^{2} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}}\right)\right) \]
   8. Step-by-step derivation

      1. frac-2neg 71.1%

         \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.25}\right)}^{2} \cdot \left(\sqrt{\color{blue}{\frac{-d}{-\ell}}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}\right)\right) \]

      2. sqrt-div 77.1%

         \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.25}\right)}^{2} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}\right)\right) \]

   9. Applied egg-rr 77.1%

      \[\leadsto {\left({\left(\frac{d}{h}\right)}^{0.25}\right)}^{2} \cdot \left(\color{blue}{\frac{\sqrt{-d}}{\sqrt{-\ell}}} \cdot \left(1 - 0.5 \cdot \frac{{\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot h}{\ell}\right)\right) \]

   if -4.999999999999985e-310 < l

   1. Initial program 66.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 65.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Step-by-step derivation

      1. sqrt-div 67.4%

         \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \]

   5. Applied egg-rr 67.4%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-div 81.5%

         \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \]

   7. Applied egg-rr 81.5%

      \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-310}:\\ \;\;\;\;{\left({\left(\frac{d}{h}\right)}^{0.25}\right)}^{2} \cdot \left(\frac{\sqrt{-d}}{\sqrt{-\ell}} \cdot \left(1 - 0.5 \cdot \frac{h \cdot {\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2}}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \end{array} \]

Alternative 2: 75.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ t_1 := \sqrt{\ell \cdot h}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;\left|\frac{d}{t_1} \cdot \mathsf{fma}\left(0.5, \frac{h}{\ell} \cdot \left(-{\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right), 1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(d \cdot \frac{1}{t_1}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0))))))
        (t_1 (sqrt (* l h))))
   (if (<= t_0 2e+287)
     (*
      (sqrt (/ d l))
      (*
       (sqrt (/ d h))
       (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (/ (* M D) 2.0) d) 2.0))))))
     (if (<= t_0 INFINITY)
       (fabs
        (*
         (/ d t_1)
         (fma 0.5 (* (/ h l) (- (pow (* M (/ (* 0.5 D) d)) 2.0))) 1.0)))
       (sqrt (pow (* d (/ 1.0 t_1)) 2.0))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D) / (d * 2.0)), 2.0))));
	double t_1 = sqrt((l * h));
	double tmp;
	if (t_0 <= 2e+287) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * pow((((M * D) / 2.0) / d), 2.0)))));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fabs(((d / t_1) * fma(0.5, ((h / l) * -pow((M * ((0.5 * D) / d)), 2.0)), 1.0)));
	} else {
		tmp = sqrt(pow((d * (1.0 / t_1)), 2.0));
	}
	return tmp;
}

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)))))
	t_1 = sqrt(Float64(l * h))
	tmp = 0.0
	if (t_0 <= 2e+287)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(Float64(M * D) / 2.0) / d) ^ 2.0))))));
	elseif (t_0 <= Inf)
		tmp = abs(Float64(Float64(d / t_1) * fma(0.5, Float64(Float64(h / l) * Float64(-(Float64(M * Float64(Float64(0.5 * D) / d)) ^ 2.0))), 1.0)));
	else
		tmp = sqrt((Float64(d * Float64(1.0 / t_1)) ^ 2.0));
	end
	return tmp
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 2e+287], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(N[(M * D), $MachinePrecision] / 2.0), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[Abs[N[(N[(d / t$95$1), $MachinePrecision] * N[(0.5 * N[(N[(h / l), $MachinePrecision] * (-N[Power[N[(M * N[(N[(0.5 * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Power[N[(d * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 2.0000000000000002e287

   1. Initial program 87.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Step-by-step derivation

      1. frac-times 87.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot -0.5\right)\right)\right) \]

      2. associate-/r* 87.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{2} \cdot -0.5\right)\right)\right) \]

   5. Applied egg-rr 87.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{2} \cdot -0.5\right)\right)\right) \]

   if 2.0000000000000002e287 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < +inf.0

   1. Initial program 56.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}} \cdot \sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}}\right) \]

      2. pow2 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\color{blue}{\left(\sqrt{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. *-commutative 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{1}{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. sqrt-prod 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{1}{2}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. sqrt-pow1 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. pow1 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. div-inv 56.4%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. metadata-eval 56.4%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. metadata-eval 56.4%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\color{blue}{0.5}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   3. Applied egg-rr 56.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   4. Step-by-step derivation

      1. expm1-log1p-u 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. expm1-udef 56.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. metadata-eval 56.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. unpow1/2 56.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{\ell}}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. Applied egg-rr 56.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]
   6. Step-by-step derivation

      1. expm1-def 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. expm1-log1p 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. Simplified 56.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. Applied egg-rr 59.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 59.4%

          \[\leadsto \sqrt{\color{blue}{\left(\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right) \cdot \left(\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right)}} \]

       2. rem-sqrt-square 99.8%

          \[\leadsto \color{blue}{\left|\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right|} \]

       3. *-commutative 99.8%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}\right| \]

       4. cancel-sign-sub-inv 99.8%

          \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 + \left(-\frac{h}{\ell}\right) \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}\right| \]

       5. *-commutative 99.8%

          \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)}\right)\right| \]

       6. +-commutative 99.8%

          \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right) + 1\right)}\right| \]

       7. associate-*l* 99.8%

          \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\color{blue}{0.5 \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-\frac{h}{\ell}\right)\right)} + 1\right)\right| \]

       8. fma-def 99.8%

          \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(0.5, {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-\frac{h}{\ell}\right), 1\right)}\right| \]

   11. Simplified 99.8%

       \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(0.5, {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2} \cdot \frac{-h}{\ell}, 1\right)\right|} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 0.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 17.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 17.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 17.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 17.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 16.3%

         \[\leadsto \color{blue}{\sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot \sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

      2. sqrt-unprod 21.4%

         \[\leadsto \color{blue}{\sqrt{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

      3. pow2 21.4%

         \[\leadsto \sqrt{\color{blue}{{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}^{2}}} \]

      4. associate-/l/ 21.4%

         \[\leadsto \sqrt{{\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)}^{2}} \]

      5. sqrt-div 21.3%

         \[\leadsto \sqrt{{\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)}^{2}} \]

      6. metadata-eval 21.3%

         \[\leadsto \sqrt{{\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)}^{2}} \]

   8. Applied egg-rr 21.3%

      \[\leadsto \color{blue}{\sqrt{{\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)}^{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(0.5, \frac{h}{\ell} \cdot \left(-{\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right), 1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(d \cdot \frac{1}{\sqrt{\ell \cdot h}}\right)}^{2}}\\ \end{array} \]

Alternative 3: 76.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right)\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\left|\frac{d}{t_0} \cdot \mathsf{fma}\left(0.5, \frac{h}{\ell} \cdot \left(-{\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right), 1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(d \cdot \frac{1}{t_0}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))
        (t_1
         (*
          (* (pow (/ d h) 0.5) (pow (/ d l) 0.5))
          (- 1.0 (* (/ h l) (* 0.5 (pow (/ (* M D) (* d 2.0)) 2.0)))))))
   (if (<= t_1 2e+287)
     (*
      (sqrt (/ d h))
      (*
       (sqrt (/ d l))
       (- 1.0 (* 0.5 (pow (* (* (* 0.5 M) (/ D d)) (sqrt (/ h l))) 2.0)))))
     (if (<= t_1 INFINITY)
       (fabs
        (*
         (/ d t_0)
         (fma 0.5 (* (/ h l) (- (pow (* M (/ (* 0.5 D) d)) 2.0))) 1.0)))
       (sqrt (pow (* d (/ 1.0 t_0)) 2.0))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double t_1 = (pow((d / h), 0.5) * pow((d / l), 0.5)) * (1.0 - ((h / l) * (0.5 * pow(((M * D) / (d * 2.0)), 2.0))));
	double tmp;
	if (t_1 <= 2e+287) {
		tmp = sqrt((d / h)) * (sqrt((d / l)) * (1.0 - (0.5 * pow((((0.5 * M) * (D / d)) * sqrt((h / l))), 2.0))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fabs(((d / t_0) * fma(0.5, ((h / l) * -pow((M * ((0.5 * D) / d)), 2.0)), 1.0)));
	} else {
		tmp = sqrt(pow((d * (1.0 / t_0)), 2.0));
	}
	return tmp;
}

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(Float64((Float64(d / h) ^ 0.5) * (Float64(d / l) ^ 0.5)) * Float64(1.0 - Float64(Float64(h / l) * Float64(0.5 * (Float64(Float64(M * D) / Float64(d * 2.0)) ^ 2.0)))))
	tmp = 0.0
	if (t_1 <= 2e+287)
		tmp = Float64(sqrt(Float64(d / h)) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * (Float64(Float64(Float64(0.5 * M) * Float64(D / d)) * sqrt(Float64(h / l))) ^ 2.0)))));
	elseif (t_1 <= Inf)
		tmp = abs(Float64(Float64(d / t_0) * fma(0.5, Float64(Float64(h / l) * Float64(-(Float64(M * Float64(Float64(0.5 * D) / d)) ^ 2.0))), 1.0)));
	else
		tmp = sqrt((Float64(d * Float64(1.0 / t_0)) ^ 2.0));
	end
	return tmp
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(d / h), $MachinePrecision], 0.5], $MachinePrecision] * N[Power[N[(d / l), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(h / l), $MachinePrecision] * N[(0.5 * N[Power[N[(N[(M * D), $MachinePrecision] / N[(d * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+287], N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[Power[N[(N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(h / l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Abs[N[(N[(d / t$95$0), $MachinePrecision] * N[(0.5 * N[(N[(h / l), $MachinePrecision] * (-N[Power[N[(M * N[(N[(0.5 * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[Power[N[(d * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < 2.0000000000000002e287

   1. Initial program 87.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 86.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}} \cdot \sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}\right)\right) \]

      2. pow2 86.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\right)}^{2}}\right)\right) \]

      3. sqrt-prod 86.3%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right)\right) \]

      4. sqrt-pow1 88.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]

      5. metadata-eval 88.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]

      6. pow1 88.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]

      7. div-inv 88.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]

      8. metadata-eval 88.4%

         \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right) \]

   5. Applied egg-rr 88.4%

      \[\leadsto \sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \color{blue}{{\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right)\right) \]

   if 2.0000000000000002e287 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l)))) < +inf.0

   1. Initial program 56.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}} \cdot \sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}}\right) \]

      2. pow2 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\color{blue}{\left(\sqrt{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. *-commutative 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{1}{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. sqrt-prod 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{1}{2}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. sqrt-pow1 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. pow1 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. div-inv 56.4%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. metadata-eval 56.4%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. metadata-eval 56.4%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\color{blue}{0.5}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   3. Applied egg-rr 56.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   4. Step-by-step derivation

      1. expm1-log1p-u 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. expm1-udef 56.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. metadata-eval 56.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. unpow1/2 56.1%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{\ell}}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. Applied egg-rr 56.1%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]
   6. Step-by-step derivation

      1. expm1-def 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. expm1-log1p 56.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. Simplified 56.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. Applied egg-rr 59.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 59.4%

          \[\leadsto \sqrt{\color{blue}{\left(\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right) \cdot \left(\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right)}} \]

       2. rem-sqrt-square 99.8%

          \[\leadsto \color{blue}{\left|\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}\right|} \]

       3. *-commutative 99.8%

          \[\leadsto \left|\color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}\right| \]

       4. cancel-sign-sub-inv 99.8%

          \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 + \left(-\frac{h}{\ell}\right) \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)}\right| \]

       5. *-commutative 99.8%

          \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)}\right)\right| \]

       6. +-commutative 99.8%

          \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(\left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right) + 1\right)}\right| \]

       7. associate-*l* 99.8%

          \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(\color{blue}{0.5 \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-\frac{h}{\ell}\right)\right)} + 1\right)\right| \]

       8. fma-def 99.8%

          \[\leadsto \left|\frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\mathsf{fma}\left(0.5, {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \left(-\frac{h}{\ell}\right), 1\right)}\right| \]

   11. Simplified 99.8%

       \[\leadsto \color{blue}{\left|\frac{d}{\sqrt{h \cdot \ell}} \cdot \mathsf{fma}\left(0.5, {\left(M \cdot \frac{D \cdot 0.5}{d}\right)}^{2} \cdot \frac{-h}{\ell}, 1\right)\right|} \]

   if +inf.0 < (*.f64 (*.f64 (pow.f64 (/.f64 d h) (/.f64 1 2)) (pow.f64 (/.f64 d l) (/.f64 1 2))) (-.f64 1 (*.f64 (*.f64 (/.f64 1 2) (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)) (/.f64 h l))))

   1. Initial program 0.0%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 17.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 17.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 17.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 17.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 16.3%

         \[\leadsto \color{blue}{\sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \cdot \sqrt{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}}} \]

      2. sqrt-unprod 21.4%

         \[\leadsto \color{blue}{\sqrt{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right) \cdot \left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}} \]

      3. pow2 21.4%

         \[\leadsto \sqrt{\color{blue}{{\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)}^{2}}} \]

      4. associate-/l/ 21.4%

         \[\leadsto \sqrt{{\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)}^{2}} \]

      5. sqrt-div 21.3%

         \[\leadsto \sqrt{{\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)}^{2}} \]

      6. metadata-eval 21.3%

         \[\leadsto \sqrt{{\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)}^{2}} \]

   8. Applied egg-rr 21.3%

      \[\leadsto \color{blue}{\sqrt{{\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)}^{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot {\left(\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right)\right)\\ \mathbf{elif}\;\left({\left(\frac{d}{h}\right)}^{0.5} \cdot {\left(\frac{d}{\ell}\right)}^{0.5}\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\left|\frac{d}{\sqrt{\ell \cdot h}} \cdot \mathsf{fma}\left(0.5, \frac{h}{\ell} \cdot \left(-{\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right), 1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(d \cdot \frac{1}{\sqrt{\ell \cdot h}}\right)}^{2}}\\ \end{array} \]

Alternative 4: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot t_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(t_0 \cdot -0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (* (/ D d) (/ M 2.0)) 2.0)))
   (if (<= d -9.5e-300)
     (*
      (/ (sqrt (- d)) (sqrt (- h)))
      (* (sqrt (/ d l)) (- 1.0 (* 0.5 (* (/ h l) t_0)))))
     (*
      (/ (sqrt d) (sqrt l))
      (* (/ (sqrt d) (sqrt h)) (+ 1.0 (* (/ h l) (* t_0 -0.5))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow(((D / d) * (M / 2.0)), 2.0);
	double tmp;
	if (d <= -9.5e-300) {
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * t_0))));
	} else {
		tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * (1.0 + ((h / l) * (t_0 * -0.5))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((d_1 / d) * (m / 2.0d0)) ** 2.0d0
    if (d <= (-9.5d-300)) then
        tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h / l) * t_0))))
    else
        tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * (1.0d0 + ((h / l) * (t_0 * (-0.5d0)))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow(((D / d) * (M / 2.0)), 2.0);
	double tmp;
	if (d <= -9.5e-300) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * (Math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * t_0))));
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(l)) * ((Math.sqrt(d) / Math.sqrt(h)) * (1.0 + ((h / l) * (t_0 * -0.5))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.pow(((D / d) * (M / 2.0)), 2.0)
	tmp = 0
	if d <= -9.5e-300:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * (math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * t_0))))
	else:
		tmp = (math.sqrt(d) / math.sqrt(l)) * ((math.sqrt(d) / math.sqrt(h)) * (1.0 + ((h / l) * (t_0 * -0.5))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0
	tmp = 0.0
	if (d <= -9.5e-300)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * t_0)))));
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(l)) * Float64(Float64(sqrt(d) / sqrt(h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(t_0 * -0.5)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = ((D / d) * (M / 2.0)) ^ 2.0;
	tmp = 0.0;
	if (d <= -9.5e-300)
		tmp = (sqrt(-d) / sqrt(-h)) * (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * t_0))));
	else
		tmp = (sqrt(d) / sqrt(l)) * ((sqrt(d) / sqrt(h)) * (1.0 + ((h / l) * (t_0 * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[d, -9.5e-300], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[l], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -9.5000000000000007e-300

   1. Initial program 71.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. frac-2neg 70.3%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. sqrt-div 76.5%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   if -9.5000000000000007e-300 < d

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.5%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Step-by-step derivation

      1. sqrt-div 66.4%

         \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \]

   5. Applied egg-rr 66.4%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \]
   6. Step-by-step derivation

      1. sqrt-div 80.3%

         \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \]

   7. Applied egg-rr 80.3%

      \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot -0.5\right)\right)\right)\\ \end{array} \]

Alternative 5: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0
         (*
          (sqrt (/ d l))
          (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0)))))))
   (if (<= d -9.5e-300)
     (* (/ (sqrt (- d)) (sqrt (- h))) t_0)
     (* (/ (sqrt d) (sqrt h)) t_0))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M / 2.0)), 2.0))));
	double tmp;
	if (d <= -9.5e-300) {
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	} else {
		tmp = (sqrt(d) / sqrt(h)) * t_0;
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0))))
    if (d <= (-9.5d-300)) then
        tmp = (sqrt(-d) / sqrt(-h)) * t_0
    else
        tmp = (sqrt(d) / sqrt(h)) * t_0
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0))));
	double tmp;
	if (d <= -9.5e-300) {
		tmp = (Math.sqrt(-d) / Math.sqrt(-h)) * t_0;
	} else {
		tmp = (Math.sqrt(d) / Math.sqrt(h)) * t_0;
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0))))
	tmp = 0
	if d <= -9.5e-300:
		tmp = (math.sqrt(-d) / math.sqrt(-h)) * t_0
	else:
		tmp = (math.sqrt(d) / math.sqrt(h)) * t_0
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0)))))
	tmp = 0.0
	if (d <= -9.5e-300)
		tmp = Float64(Float64(sqrt(Float64(-d)) / sqrt(Float64(-h))) * t_0);
	else
		tmp = Float64(Float64(sqrt(d) / sqrt(h)) * t_0);
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	t_0 = sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0))));
	tmp = 0.0;
	if (d <= -9.5e-300)
		tmp = (sqrt(-d) / sqrt(-h)) * t_0;
	else
		tmp = (sqrt(d) / sqrt(h)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -9.5e-300], N[(N[(N[Sqrt[(-d)], $MachinePrecision] / N[Sqrt[(-h)], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[Sqrt[d], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -9.5000000000000007e-300

   1. Initial program 71.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. frac-2neg 70.3%

         \[\leadsto \sqrt{\color{blue}{\frac{-d}{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

      2. sqrt-div 76.5%

         \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   5. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{\frac{\sqrt{-d}}{\sqrt{-h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]

   if -9.5000000000000007e-300 < d

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 64.6%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. sqrt-div 80.3%

         \[\leadsto \frac{\sqrt{d}}{\sqrt{\ell}} \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right) \]

   5. Applied egg-rr 76.2%

      \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{-300}:\\ \;\;\;\;\frac{\sqrt{-d}}{\sqrt{-h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{d}}{\sqrt{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 6: 59.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := {\left(\frac{d}{h}\right)}^{0.25}\\ t_2 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -1.9 \cdot 10^{+125}:\\ \;\;\;\;t_1 \cdot \left(t_1 \cdot t_2\right)\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{t_0}\right)\right)\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-17}:\\ \;\;\;\;t_2 \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-299}:\\ \;\;\;\;\frac{d}{e^{\mathsf{log1p}\left(t_0\right)} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h))) (t_1 (pow (/ d h) 0.25)) (t_2 (sqrt (/ d l))))
   (if (<= d -1.9e+125)
     (* t_1 (* t_1 t_2))
     (if (<= d -1.85e+73)
       (log1p (expm1 (/ d t_0)))
       (if (<= d -9.2e-17)
         (* t_2 (sqrt (/ d h)))
         (if (<= d 4.05e-299)
           (/ d (+ (exp (log1p t_0)) -1.0))
           (*
            (/ d (* (sqrt l) (sqrt h)))
            (+ 1.0 (* -0.5 (* (/ h l) (pow (* M (* 0.5 (/ D d))) 2.0)))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double t_1 = pow((d / h), 0.25);
	double t_2 = sqrt((d / l));
	double tmp;
	if (d <= -1.9e+125) {
		tmp = t_1 * (t_1 * t_2);
	} else if (d <= -1.85e+73) {
		tmp = log1p(expm1((d / t_0)));
	} else if (d <= -9.2e-17) {
		tmp = t_2 * sqrt((d / h));
	} else if (d <= 4.05e-299) {
		tmp = d / (exp(log1p(t_0)) + -1.0);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (-0.5 * ((h / l) * pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((l * h));
	double t_1 = Math.pow((d / h), 0.25);
	double t_2 = Math.sqrt((d / l));
	double tmp;
	if (d <= -1.9e+125) {
		tmp = t_1 * (t_1 * t_2);
	} else if (d <= -1.85e+73) {
		tmp = Math.log1p(Math.expm1((d / t_0)));
	} else if (d <= -9.2e-17) {
		tmp = t_2 * Math.sqrt((d / h));
	} else if (d <= 4.05e-299) {
		tmp = d / (Math.exp(Math.log1p(t_0)) + -1.0);
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + (-0.5 * ((h / l) * Math.pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((l * h))
	t_1 = math.pow((d / h), 0.25)
	t_2 = math.sqrt((d / l))
	tmp = 0
	if d <= -1.9e+125:
		tmp = t_1 * (t_1 * t_2)
	elif d <= -1.85e+73:
		tmp = math.log1p(math.expm1((d / t_0)))
	elif d <= -9.2e-17:
		tmp = t_2 * math.sqrt((d / h))
	elif d <= 4.05e-299:
		tmp = d / (math.exp(math.log1p(t_0)) + -1.0)
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + (-0.5 * ((h / l) * math.pow((M * (0.5 * (D / d))), 2.0))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(d / h) ^ 0.25
	t_2 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -1.9e+125)
		tmp = Float64(t_1 * Float64(t_1 * t_2));
	elseif (d <= -1.85e+73)
		tmp = log1p(expm1(Float64(d / t_0)));
	elseif (d <= -9.2e-17)
		tmp = Float64(t_2 * sqrt(Float64(d / h)));
	elseif (d <= 4.05e-299)
		tmp = Float64(d / Float64(exp(log1p(t_0)) + -1.0));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(d / h), $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -1.9e+125], N[(t$95$1 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.85e+73], N[Log[1 + N[(Exp[N[(d / t$95$0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -9.2e-17], N[(t$95$2 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.05e-299], N[(d / N[(N[Exp[N[Log[1 + t$95$0], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.90000000000000001e125

   1. Initial program 75.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 75.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 4.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 4.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 4.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 4.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 0.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 0.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 0.5%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 0.5%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 0.5%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 0.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 0.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 4.0%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 4.0%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 4.0%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 4.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. unpow1/2 0.0%

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. metadata-eval 0.0%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. sqrt-div 62.9%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. *-commutative 62.9%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

       9. add-sqr-sqrt 62.9%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\sqrt{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \sqrt{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right)} \]

       10. associate-*r* 62.9%

           \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \sqrt{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}} \]

       11. sqrt-pow1 63.0%

           \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}\right) \cdot \sqrt{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

       12. metadata-eval 63.0%

           \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)}\right) \cdot \sqrt{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

       13. metadata-eval 63.0%

           \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.25}}\right) \cdot \sqrt{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

       14. sqrt-pow1 62.9%

           \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right) \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}} \]

       15. metadata-eval 62.9%

           \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)} \]

       16. metadata-eval 62.9%

           \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right) \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.25}} \]

   12. Applied egg-rr 62.9%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right) \cdot {\left(\frac{d}{h}\right)}^{0.25}} \]

   if -1.90000000000000001e125 < d < -1.84999999999999987e73

   1. Initial program 80.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 9.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 9.6%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 9.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 9.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 0.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 0.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 0.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 0.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 9.6%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 9.6%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 9.6%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 9.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. log1p-expm1-u 54.1%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

   12. Applied egg-rr 54.1%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

   if -1.84999999999999987e73 < d < -9.20000000000000035e-17

   1. Initial program 81.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 81.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 3.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 3.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 3.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 3.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 2.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 3.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 3.0%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 3.0%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 3.0%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 3.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 2.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 3.2%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 3.2%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 3.2%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 3.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. unpow1/2 0.0%

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. metadata-eval 0.0%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. sqrt-div 64.5%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. *-commutative 64.5%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

       9. metadata-eval 64.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \]

       10. unpow1/2 64.5%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

   12. Applied egg-rr 64.5%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   if -9.20000000000000035e-17 < d < 4.0499999999999998e-299

   1. Initial program 58.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 58.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 19.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 19.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 19.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 19.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 6.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 6.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 6.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 6.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 6.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 19.2%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 19.2%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 19.2%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 19.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 19.2%

          \[\leadsto \frac{d}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{h \cdot \ell}\right)\right)}} \]

       2. expm1-udef 40.7%

          \[\leadsto \frac{d}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{h \cdot \ell}\right)} - 1}} \]

   12. Applied egg-rr 40.7%

       \[\leadsto \frac{d}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{h \cdot \ell}\right)} - 1}} \]

   if 4.0499999999999998e-299 < d

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 29.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      2. expm1-udef 20.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1} \]

   3. Applied egg-rr 26.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 36.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-log1p 78.2%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

      3. cancel-sign-sub-inv 78.2%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

      4. metadata-eval 78.2%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. associate-*l* 78.2%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left({\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   5. Simplified 78.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+125}:\\ \;\;\;\;{\left(\frac{d}{h}\right)}^{0.25} \cdot \left({\left(\frac{d}{h}\right)}^{0.25} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-299}:\\ \;\;\;\;\frac{d}{e^{\mathsf{log1p}\left(\sqrt{\ell \cdot h}\right)} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 7: 64.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := {\left(\frac{d}{\ell}\right)}^{0.25}\\ \mathbf{if}\;d \leq -7.5 \cdot 10^{+170}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{\frac{d}{h}} \cdot t_0\right)\\ \mathbf{elif}\;d \leq -2.25 \cdot 10^{-104}:\\ \;\;\;\;\left(1 - {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{{d}^{2}}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-299}:\\ \;\;\;\;\frac{d}{e^{\mathsf{log1p}\left(\sqrt{\ell \cdot h}\right)} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (pow (/ d l) 0.25)))
   (if (<= d -7.5e+170)
     (* t_0 (* (sqrt (/ d h)) t_0))
     (if (<= d -2.25e-104)
       (*
        (- 1.0 (* (pow (* (/ D d) (/ M 2.0)) 2.0) (* 0.5 (/ h l))))
        (sqrt (/ (pow d 2.0) (* l h))))
       (if (<= d 4.05e-299)
         (/ d (+ (exp (log1p (sqrt (* l h)))) -1.0))
         (*
          (/ d (* (sqrt l) (sqrt h)))
          (+ 1.0 (* -0.5 (* (/ h l) (pow (* M (* 0.5 (/ D d))) 2.0))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = pow((d / l), 0.25);
	double tmp;
	if (d <= -7.5e+170) {
		tmp = t_0 * (sqrt((d / h)) * t_0);
	} else if (d <= -2.25e-104) {
		tmp = (1.0 - (pow(((D / d) * (M / 2.0)), 2.0) * (0.5 * (h / l)))) * sqrt((pow(d, 2.0) / (l * h)));
	} else if (d <= 4.05e-299) {
		tmp = d / (exp(log1p(sqrt((l * h)))) + -1.0);
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (-0.5 * ((h / l) * pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.pow((d / l), 0.25);
	double tmp;
	if (d <= -7.5e+170) {
		tmp = t_0 * (Math.sqrt((d / h)) * t_0);
	} else if (d <= -2.25e-104) {
		tmp = (1.0 - (Math.pow(((D / d) * (M / 2.0)), 2.0) * (0.5 * (h / l)))) * Math.sqrt((Math.pow(d, 2.0) / (l * h)));
	} else if (d <= 4.05e-299) {
		tmp = d / (Math.exp(Math.log1p(Math.sqrt((l * h)))) + -1.0);
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + (-0.5 * ((h / l) * Math.pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.pow((d / l), 0.25)
	tmp = 0
	if d <= -7.5e+170:
		tmp = t_0 * (math.sqrt((d / h)) * t_0)
	elif d <= -2.25e-104:
		tmp = (1.0 - (math.pow(((D / d) * (M / 2.0)), 2.0) * (0.5 * (h / l)))) * math.sqrt((math.pow(d, 2.0) / (l * h)))
	elif d <= 4.05e-299:
		tmp = d / (math.exp(math.log1p(math.sqrt((l * h)))) + -1.0)
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + (-0.5 * ((h / l) * math.pow((M * (0.5 * (D / d))), 2.0))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(d / l) ^ 0.25
	tmp = 0.0
	if (d <= -7.5e+170)
		tmp = Float64(t_0 * Float64(sqrt(Float64(d / h)) * t_0));
	elseif (d <= -2.25e-104)
		tmp = Float64(Float64(1.0 - Float64((Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0) * Float64(0.5 * Float64(h / l)))) * sqrt(Float64((d ^ 2.0) / Float64(l * h))));
	elseif (d <= 4.05e-299)
		tmp = Float64(d / Float64(exp(log1p(sqrt(Float64(l * h)))) + -1.0));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Power[N[(d / l), $MachinePrecision], 0.25], $MachinePrecision]}, If[LessEqual[d, -7.5e+170], N[(t$95$0 * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.25e-104], N[(N[(1.0 - N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Power[d, 2.0], $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.05e-299], N[(d / N[(N[Exp[N[Log[1 + N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.5000000000000002e170

   1. Initial program 79.7%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 79.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 1.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 1.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 1.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 1.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 0.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 0.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 0.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 0.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 0.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 0.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 0.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 1.2%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 1.2%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 1.2%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 1.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. unpow1/2 0.0%

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. metadata-eval 0.0%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. sqrt-div 72.8%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. add-sqr-sqrt 72.7%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{d}{\ell}}} \cdot \sqrt{\sqrt{\frac{d}{\ell}}}\right)} \]

       9. associate-*r* 72.9%

          \[\leadsto \color{blue}{\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \sqrt{\sqrt{\frac{d}{\ell}}}\right) \cdot \sqrt{\sqrt{\frac{d}{\ell}}}} \]

       10. metadata-eval 72.9%

           \[\leadsto \left({\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \cdot \sqrt{\sqrt{\frac{d}{\ell}}}\right) \cdot \sqrt{\sqrt{\frac{d}{\ell}}} \]

       11. unpow1/2 72.9%

           \[\leadsto \left(\color{blue}{\sqrt{\frac{d}{h}}} \cdot \sqrt{\sqrt{\frac{d}{\ell}}}\right) \cdot \sqrt{\sqrt{\frac{d}{\ell}}} \]

       12. pow1/2 72.9%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}}\right) \cdot \sqrt{\sqrt{\frac{d}{\ell}}} \]

       13. metadata-eval 72.9%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \sqrt{{\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}}\right) \cdot \sqrt{\sqrt{\frac{d}{\ell}}} \]

       14. sqrt-pow1 72.9%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}\right) \cdot \sqrt{\sqrt{\frac{d}{\ell}}} \]

       15. metadata-eval 72.9%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)}\right) \cdot \sqrt{\sqrt{\frac{d}{\ell}}} \]

       16. metadata-eval 72.9%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{0.25}}\right) \cdot \sqrt{\sqrt{\frac{d}{\ell}}} \]

       17. pow1/2 72.9%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right) \cdot \sqrt{\color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}}} \]

       18. metadata-eval 72.9%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right) \cdot \sqrt{{\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}} \]

       19. sqrt-pow1 72.9%

           \[\leadsto \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right) \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}} \]

   12. Applied egg-rr 72.9%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right) \cdot {\left(\frac{d}{\ell}\right)}^{0.25}} \]

   if -7.5000000000000002e170 < d < -2.2499999999999999e-104

   1. Initial program 77.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 77.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}} \cdot \sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}}\right) \]

      2. pow2 77.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 77.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\color{blue}{\left(\sqrt{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. *-commutative 77.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 77.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{1}{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. sqrt-prod 77.7%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{1}{2}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. sqrt-pow1 80.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 80.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. pow1 80.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. div-inv 80.9%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. metadata-eval 80.9%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. metadata-eval 80.9%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\color{blue}{0.5}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   3. Applied egg-rr 80.9%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]

   5. Applied egg-rr 12.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 23.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 66.8%

         \[\leadsto \color{blue}{\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right)\right) \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

      3. *-commutative 66.8%

         \[\leadsto \left(1 - \color{blue}{\left(0.5 \cdot {\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right) \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      4. *-commutative 66.8%

         \[\leadsto \left(1 - \color{blue}{\left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot 0.5\right)} \cdot \frac{h}{\ell}\right) \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      5. associate-*l* 66.8%

         \[\leadsto \left(1 - \color{blue}{{\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)}\right) \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

      6. times-frac 66.7%

         \[\leadsto \left(1 - {\color{blue}{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}} \]

   7. Simplified 66.7%

      \[\leadsto \color{blue}{\left(1 - {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   if -2.2499999999999999e-104 < d < 4.0499999999999998e-299

   1. Initial program 47.1%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 47.1%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 20.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 20.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 20.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 20.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 9.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 9.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 9.4%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 9.4%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 9.4%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 9.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 9.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 20.1%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 20.1%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 20.1%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 20.1%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 20.1%

          \[\leadsto \frac{d}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{h \cdot \ell}\right)\right)}} \]

       2. expm1-udef 42.8%

          \[\leadsto \frac{d}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{h \cdot \ell}\right)} - 1}} \]

   12. Applied egg-rr 42.8%

       \[\leadsto \frac{d}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{h \cdot \ell}\right)} - 1}} \]

   if 4.0499999999999998e-299 < d

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 29.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      2. expm1-udef 20.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1} \]

   3. Applied egg-rr 26.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 36.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-log1p 78.2%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

      3. cancel-sign-sub-inv 78.2%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

      4. metadata-eval 78.2%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. associate-*l* 78.2%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left({\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   5. Simplified 78.2%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{+170}:\\ \;\;\;\;{\left(\frac{d}{\ell}\right)}^{0.25} \cdot \left(\sqrt{\frac{d}{h}} \cdot {\left(\frac{d}{\ell}\right)}^{0.25}\right)\\ \mathbf{elif}\;d \leq -2.25 \cdot 10^{-104}:\\ \;\;\;\;\left(1 - {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \left(0.5 \cdot \frac{h}{\ell}\right)\right) \cdot \sqrt{\frac{{d}^{2}}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-299}:\\ \;\;\;\;\frac{d}{e^{\mathsf{log1p}\left(\sqrt{\ell \cdot h}\right)} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 8: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{-300}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -9.5e-300)
   (*
    (*
     (sqrt (/ d l))
     (- 1.0 (* 0.5 (* (/ h l) (pow (* (/ D d) (/ M 2.0)) 2.0)))))
    (sqrt (/ d h)))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* M (* 0.5 (/ D d))) 2.0)))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -9.5e-300) {
		tmp = (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * pow(((D / d) * (M / 2.0)), 2.0))))) * sqrt((d / h));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (-0.5 * ((h / l) * pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-9.5d-300)) then
        tmp = (sqrt((d / l)) * (1.0d0 - (0.5d0 * ((h / l) * (((d_1 / d) * (m / 2.0d0)) ** 2.0d0))))) * sqrt((d / h))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -9.5e-300) {
		tmp = (Math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * Math.pow(((D / d) * (M / 2.0)), 2.0))))) * Math.sqrt((d / h));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + (-0.5 * ((h / l) * Math.pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -9.5e-300:
		tmp = (math.sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * math.pow(((D / d) * (M / 2.0)), 2.0))))) * math.sqrt((d / h))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + (-0.5 * ((h / l) * math.pow((M * (0.5 * (D / d))), 2.0))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -9.5e-300)
		tmp = Float64(Float64(sqrt(Float64(d / l)) * Float64(1.0 - Float64(0.5 * Float64(Float64(h / l) * (Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0))))) * sqrt(Float64(d / h)));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -9.5e-300)
		tmp = (sqrt((d / l)) * (1.0 - (0.5 * ((h / l) * (((D / d) * (M / 2.0)) ^ 2.0))))) * sqrt((d / h));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (-0.5 * ((h / l) * ((M * (0.5 * (D / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -9.5e-300], N[(N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(1.0 - N[(0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -9.5000000000000007e-300

   1. Initial program 71.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} \]

   if -9.5000000000000007e-300 < d

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 28.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      2. expm1-udef 20.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1} \]

   3. Applied egg-rr 25.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 35.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-log1p 75.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

      3. cancel-sign-sub-inv 75.8%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

      4. metadata-eval 75.8%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. associate-*l* 75.8%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left({\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   5. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{-300}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} \cdot \left(1 - 0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2}\right)\right)\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 9: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -9.5e-300)
   (*
    (sqrt (/ d l))
    (*
     (+ 1.0 (* (/ h l) (* (pow (* (/ D d) (/ M 2.0)) 2.0) -0.5)))
     (sqrt (/ d h))))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* M (* 0.5 (/ D d))) 2.0)))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -9.5e-300) {
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * (pow(((D / d) * (M / 2.0)), 2.0) * -0.5))) * sqrt((d / h)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (-0.5 * ((h / l) * pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-9.5d-300)) then
        tmp = sqrt((d / l)) * ((1.0d0 + ((h / l) * ((((d_1 / d) * (m / 2.0d0)) ** 2.0d0) * (-0.5d0)))) * sqrt((d / h)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -9.5e-300) {
		tmp = Math.sqrt((d / l)) * ((1.0 + ((h / l) * (Math.pow(((D / d) * (M / 2.0)), 2.0) * -0.5))) * Math.sqrt((d / h)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + (-0.5 * ((h / l) * Math.pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -9.5e-300:
		tmp = math.sqrt((d / l)) * ((1.0 + ((h / l) * (math.pow(((D / d) * (M / 2.0)), 2.0) * -0.5))) * math.sqrt((d / h)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + (-0.5 * ((h / l) * math.pow((M * (0.5 * (D / d))), 2.0))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -9.5e-300)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(Float64(1.0 + Float64(Float64(h / l) * Float64((Float64(Float64(D / d) * Float64(M / 2.0)) ^ 2.0) * -0.5))) * sqrt(Float64(d / h))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -9.5e-300)
		tmp = sqrt((d / l)) * ((1.0 + ((h / l) * ((((D / d) * (M / 2.0)) ^ 2.0) * -0.5))) * sqrt((d / h)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (-0.5 * ((h / l) * ((M * (0.5 * (D / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -9.5e-300], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(N[Power[N[(N[(D / d), $MachinePrecision] * N[(M / 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -9.5000000000000007e-300

   1. Initial program 71.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 70.3%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]

   if -9.5000000000000007e-300 < d

   1. Initial program 65.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 28.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      2. expm1-udef 20.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1} \]

   3. Applied egg-rr 25.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 35.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-log1p 75.8%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

      3. cancel-sign-sub-inv 75.8%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

      4. metadata-eval 75.8%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. associate-*l* 75.8%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left({\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   5. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot -0.5\right)\right) \cdot \sqrt{\frac{d}{h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 10: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq 1.02 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d 1.02e-231)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (* (/ h l) (* -0.5 (pow (/ (/ (* M D) 2.0) d) 2.0))))))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* M (* 0.5 (/ D d))) 2.0)))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 1.02e-231) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * pow((((M * D) / 2.0) / d), 2.0)))));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (-0.5 * ((h / l) * pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= 1.02d-231) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h / l) * ((-0.5d0) * ((((m * d_1) / 2.0d0) / d) ** 2.0d0)))))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= 1.02e-231) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * Math.pow((((M * D) / 2.0) / d), 2.0)))));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + (-0.5 * ((h / l) * Math.pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= 1.02e-231:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * math.pow((((M * D) / 2.0) / d), 2.0)))))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + (-0.5 * ((h / l) * math.pow((M * (0.5 * (D / d))), 2.0))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= 1.02e-231)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h / l) * Float64(-0.5 * (Float64(Float64(Float64(M * D) / 2.0) / d) ^ 2.0))))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= 1.02e-231)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h / l) * (-0.5 * ((((M * D) / 2.0) / d) ^ 2.0)))));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (-0.5 * ((h / l) * ((M * (0.5 * (D / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, 1.02e-231], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h / l), $MachinePrecision] * N[(-0.5 * N[Power[N[(N[(N[(M * D), $MachinePrecision] / 2.0), $MachinePrecision] / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 1.02000000000000006e-231

   1. Initial program 68.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 68.0%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Step-by-step derivation

      1. frac-times 68.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{M \cdot D}{2 \cdot d}\right)}}^{2} \cdot -0.5\right)\right)\right) \]

      2. associate-/r* 68.8%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{2} \cdot -0.5\right)\right)\right) \]

   5. Applied egg-rr 68.8%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\color{blue}{\left(\frac{\frac{M \cdot D}{2}}{d}\right)}}^{2} \cdot -0.5\right)\right)\right) \]

   if 1.02000000000000006e-231 < d

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 30.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      2. expm1-udef 23.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1} \]

   3. Applied egg-rr 29.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 39.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-log1p 79.3%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

      3. cancel-sign-sub-inv 79.3%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

      4. metadata-eval 79.3%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. associate-*l* 79.3%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left({\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   5. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 1.02 \cdot 10^{-231}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left(-0.5 \cdot {\left(\frac{\frac{M \cdot D}{2}}{d}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 11: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= l 2e-31)
   (*
    (sqrt (/ d l))
    (*
     (sqrt (/ d h))
     (+ 1.0 (/ (* h (* (pow (* (* 0.5 M) (/ D d)) 2.0) -0.5)) l))))
   (*
    (/ d (* (sqrt l) (sqrt h)))
    (+ 1.0 (* -0.5 (* (/ h l) (pow (* M (* 0.5 (/ D d))) 2.0)))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2e-31) {
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h * (pow(((0.5 * M) * (D / d)), 2.0) * -0.5)) / l)));
	} else {
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (-0.5 * ((h / l) * pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (l <= 2d-31) then
        tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0d0 + ((h * ((((0.5d0 * m) * (d_1 / d)) ** 2.0d0) * (-0.5d0))) / l)))
    else
        tmp = (d / (sqrt(l) * sqrt(h))) * (1.0d0 + ((-0.5d0) * ((h / l) * ((m * (0.5d0 * (d_1 / d))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (l <= 2e-31) {
		tmp = Math.sqrt((d / l)) * (Math.sqrt((d / h)) * (1.0 + ((h * (Math.pow(((0.5 * M) * (D / d)), 2.0) * -0.5)) / l)));
	} else {
		tmp = (d / (Math.sqrt(l) * Math.sqrt(h))) * (1.0 + (-0.5 * ((h / l) * Math.pow((M * (0.5 * (D / d))), 2.0))));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if l <= 2e-31:
		tmp = math.sqrt((d / l)) * (math.sqrt((d / h)) * (1.0 + ((h * (math.pow(((0.5 * M) * (D / d)), 2.0) * -0.5)) / l)))
	else:
		tmp = (d / (math.sqrt(l) * math.sqrt(h))) * (1.0 + (-0.5 * ((h / l) * math.pow((M * (0.5 * (D / d))), 2.0))))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (l <= 2e-31)
		tmp = Float64(sqrt(Float64(d / l)) * Float64(sqrt(Float64(d / h)) * Float64(1.0 + Float64(Float64(h * Float64((Float64(Float64(0.5 * M) * Float64(D / d)) ^ 2.0) * -0.5)) / l))));
	else
		tmp = Float64(Float64(d / Float64(sqrt(l) * sqrt(h))) * Float64(1.0 + Float64(-0.5 * Float64(Float64(h / l) * (Float64(M * Float64(0.5 * Float64(D / d))) ^ 2.0)))));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (l <= 2e-31)
		tmp = sqrt((d / l)) * (sqrt((d / h)) * (1.0 + ((h * ((((0.5 * M) * (D / d)) ^ 2.0) * -0.5)) / l)));
	else
		tmp = (d / (sqrt(l) * sqrt(h))) * (1.0 + (-0.5 * ((h / l) * ((M * (0.5 * (D / d))) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[l, 2e-31], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(h * N[(N[Power[N[(N[(0.5 * M), $MachinePrecision] * N[(D / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[(N[Sqrt[l], $MachinePrecision] * N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(N[(h / l), $MachinePrecision] * N[Power[N[(M * N[(0.5 * N[(D / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2e-31

   1. Initial program 73.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 72.2%

      \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h}{\ell} \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*l/ 75.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}}\right)\right) \]

      2. add-sqr-sqrt 46.6%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left({\color{blue}{\left(\sqrt{\frac{M}{2} \cdot \frac{D}{d}} \cdot \sqrt{\frac{M}{2} \cdot \frac{D}{d}}\right)}}^{2} \cdot -0.5\right)}{\ell}\right)\right) \]

      3. add-sqr-sqrt 75.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot -0.5\right)}{\ell}\right)\right) \]

      4. div-inv 75.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left({\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right) \]

      5. metadata-eval 75.4%

         \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left({\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right) \]

   5. Applied egg-rr 75.4%

      \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \color{blue}{\frac{h \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}}\right)\right) \]

   if 2e-31 < l

   1. Initial program 52.9%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 32.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)\right)} \]

      2. expm1-udef 17.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\right)} - 1} \]

   3. Applied egg-rr 26.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)} - 1} \]
   4. Step-by-step derivation

      1. expm1-def 45.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)\right)\right)} \]

      2. expm1-log1p 71.4%

         \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 - 0.5 \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

      3. cancel-sign-sub-inv 71.4%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \color{blue}{\left(1 + \left(-0.5\right) \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]

      4. metadata-eval 71.4%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + \color{blue}{-0.5} \cdot \left({\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}\right)\right) \]

      5. associate-*l* 71.4%

         \[\leadsto \frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left({\color{blue}{\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}}^{2} \cdot \frac{h}{\ell}\right)\right) \]

   5. Simplified 71.4%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h} \cdot \sqrt{\ell}} \cdot \left(1 + -0.5 \cdot \left({\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2} \cdot \frac{h}{\ell}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \left(1 + \frac{h \cdot \left({\left(\left(0.5 \cdot M\right) \cdot \frac{D}{d}\right)}^{2} \cdot -0.5\right)}{\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell} \cdot \sqrt{h}} \cdot \left(1 + -0.5 \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)\\ \end{array} \]

Alternative 12: 53.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \frac{d}{t_0}\\ t_2 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -3.9 \cdot 10^{+128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-299}:\\ \;\;\;\;\frac{d}{e^{\mathsf{log1p}\left(t_0\right)} + -1}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))
        (t_1 (/ d t_0))
        (t_2 (* (sqrt (/ d l)) (sqrt (/ d h)))))
   (if (<= d -3.9e+128)
     t_2
     (if (<= d -1.75e+73)
       (log1p (expm1 t_1))
       (if (<= d -3.5e-17)
         t_2
         (if (<= d 4.05e-299)
           (/ d (+ (exp (log1p t_0)) -1.0))
           (*
            t_1
            (- 1.0 (* (* 0.5 (/ h l)) (pow (* M (/ (* 0.5 D) d)) 2.0))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double t_1 = d / t_0;
	double t_2 = sqrt((d / l)) * sqrt((d / h));
	double tmp;
	if (d <= -3.9e+128) {
		tmp = t_2;
	} else if (d <= -1.75e+73) {
		tmp = log1p(expm1(t_1));
	} else if (d <= -3.5e-17) {
		tmp = t_2;
	} else if (d <= 4.05e-299) {
		tmp = d / (exp(log1p(t_0)) + -1.0);
	} else {
		tmp = t_1 * (1.0 - ((0.5 * (h / l)) * pow((M * ((0.5 * D) / d)), 2.0)));
	}
	return tmp;
}

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((l * h));
	double t_1 = d / t_0;
	double t_2 = Math.sqrt((d / l)) * Math.sqrt((d / h));
	double tmp;
	if (d <= -3.9e+128) {
		tmp = t_2;
	} else if (d <= -1.75e+73) {
		tmp = Math.log1p(Math.expm1(t_1));
	} else if (d <= -3.5e-17) {
		tmp = t_2;
	} else if (d <= 4.05e-299) {
		tmp = d / (Math.exp(Math.log1p(t_0)) + -1.0);
	} else {
		tmp = t_1 * (1.0 - ((0.5 * (h / l)) * Math.pow((M * ((0.5 * D) / d)), 2.0)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((l * h))
	t_1 = d / t_0
	t_2 = math.sqrt((d / l)) * math.sqrt((d / h))
	tmp = 0
	if d <= -3.9e+128:
		tmp = t_2
	elif d <= -1.75e+73:
		tmp = math.log1p(math.expm1(t_1))
	elif d <= -3.5e-17:
		tmp = t_2
	elif d <= 4.05e-299:
		tmp = d / (math.exp(math.log1p(t_0)) + -1.0)
	else:
		tmp = t_1 * (1.0 - ((0.5 * (h / l)) * math.pow((M * ((0.5 * D) / d)), 2.0)))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(d / t_0)
	t_2 = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)))
	tmp = 0.0
	if (d <= -3.9e+128)
		tmp = t_2;
	elseif (d <= -1.75e+73)
		tmp = log1p(expm1(t_1));
	elseif (d <= -3.5e-17)
		tmp = t_2;
	elseif (d <= 4.05e-299)
		tmp = Float64(d / Float64(exp(log1p(t_0)) + -1.0));
	else
		tmp = Float64(t_1 * Float64(1.0 - Float64(Float64(0.5 * Float64(h / l)) * (Float64(M * Float64(Float64(0.5 * D) / d)) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(d / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.9e+128], t$95$2, If[LessEqual[d, -1.75e+73], N[Log[1 + N[(Exp[t$95$1] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -3.5e-17], t$95$2, If[LessEqual[d, 4.05e-299], N[(d / N[(N[Exp[N[Log[1 + t$95$0], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(1.0 - N[(N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(M * N[(N[(0.5 * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.8999999999999997e128 or -1.75000000000000001e73 < d < -3.5000000000000002e-17

   1. Initial program 77.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 3.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 3.8%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 3.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 3.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 1.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 1.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 1.2%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 1.2%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 1.2%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 1.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 1.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 3.8%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 3.8%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 3.8%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 3.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. unpow1/2 0.0%

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. metadata-eval 0.0%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. sqrt-div 63.4%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. *-commutative 63.4%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

       9. metadata-eval 63.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \]

       10. unpow1/2 63.4%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

   12. Applied egg-rr 63.4%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   if -3.8999999999999997e128 < d < -1.75000000000000001e73

   1. Initial program 80.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 9.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 9.6%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 9.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 9.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 0.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 0.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 0.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 0.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 9.6%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 9.6%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 9.6%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 9.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. log1p-expm1-u 54.1%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

   12. Applied egg-rr 54.1%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

   if -3.5000000000000002e-17 < d < 4.0499999999999998e-299

   1. Initial program 58.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 58.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 19.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 19.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 19.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 19.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 6.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 6.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 6.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 6.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 6.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 19.2%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 19.2%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 19.2%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 19.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 19.2%

          \[\leadsto \frac{d}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{h \cdot \ell}\right)\right)}} \]

       2. expm1-udef 40.7%

          \[\leadsto \frac{d}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{h \cdot \ell}\right)} - 1}} \]

   12. Applied egg-rr 40.7%

       \[\leadsto \frac{d}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{h \cdot \ell}\right)} - 1}} \]

   if 4.0499999999999998e-299 < d

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}} \cdot \sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}}\right) \]

      2. pow2 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\color{blue}{\left(\sqrt{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. *-commutative 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 66.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{1}{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. sqrt-prod 66.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{1}{2}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. sqrt-pow1 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. pow1 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. div-inv 69.8%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. metadata-eval 69.8%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. metadata-eval 69.8%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\color{blue}{0.5}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   3. Applied egg-rr 69.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   4. Step-by-step derivation

      1. expm1-log1p-u 68.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. expm1-udef 45.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. metadata-eval 45.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. unpow1/2 45.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{\ell}}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. Applied egg-rr 45.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]
   6. Step-by-step derivation

      1. expm1-def 68.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. expm1-log1p 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. Simplified 69.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. Applied egg-rr 62.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} + \left(\left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. *-lft-identity 62.8%

          \[\leadsto \color{blue}{1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} + \left(\left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

       2. distribute-rgt-in 71.6%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)\right)} \]

       3. *-commutative 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\left(-\frac{h}{\ell}\right) \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}\right) \]

       4. cancel-sign-sub-inv 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)} \]

       5. associate-*r* 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}\right) \]

       6. associate-*r/ 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(M \cdot \color{blue}{\frac{0.5 \cdot D}{d}}\right)}^{2}\right) \]

   11. Simplified 71.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.9 \cdot 10^{+128}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-299}:\\ \;\;\;\;\frac{d}{e^{\mathsf{log1p}\left(\sqrt{\ell \cdot h}\right)} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)\\ \end{array} \]

Alternative 13: 53.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \frac{d}{t_0}\\ t_2 := {\left(\frac{d}{h}\right)}^{0.25}\\ t_3 := \sqrt{\frac{d}{\ell}}\\ \mathbf{if}\;d \leq -6 \cdot 10^{+126}:\\ \;\;\;\;t_2 \cdot \left(t_2 \cdot t_3\right)\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)\\ \mathbf{elif}\;d \leq -1.26 \cdot 10^{-17}:\\ \;\;\;\;t_3 \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-299}:\\ \;\;\;\;\frac{d}{e^{\mathsf{log1p}\left(t_0\right)} + -1}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))
        (t_1 (/ d t_0))
        (t_2 (pow (/ d h) 0.25))
        (t_3 (sqrt (/ d l))))
   (if (<= d -6e+126)
     (* t_2 (* t_2 t_3))
     (if (<= d -1.85e+73)
       (log1p (expm1 t_1))
       (if (<= d -1.26e-17)
         (* t_3 (sqrt (/ d h)))
         (if (<= d 4.05e-299)
           (/ d (+ (exp (log1p t_0)) -1.0))
           (*
            t_1
            (- 1.0 (* (* 0.5 (/ h l)) (pow (* M (/ (* 0.5 D) d)) 2.0))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double t_1 = d / t_0;
	double t_2 = pow((d / h), 0.25);
	double t_3 = sqrt((d / l));
	double tmp;
	if (d <= -6e+126) {
		tmp = t_2 * (t_2 * t_3);
	} else if (d <= -1.85e+73) {
		tmp = log1p(expm1(t_1));
	} else if (d <= -1.26e-17) {
		tmp = t_3 * sqrt((d / h));
	} else if (d <= 4.05e-299) {
		tmp = d / (exp(log1p(t_0)) + -1.0);
	} else {
		tmp = t_1 * (1.0 - ((0.5 * (h / l)) * pow((M * ((0.5 * D) / d)), 2.0)));
	}
	return tmp;
}

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((l * h));
	double t_1 = d / t_0;
	double t_2 = Math.pow((d / h), 0.25);
	double t_3 = Math.sqrt((d / l));
	double tmp;
	if (d <= -6e+126) {
		tmp = t_2 * (t_2 * t_3);
	} else if (d <= -1.85e+73) {
		tmp = Math.log1p(Math.expm1(t_1));
	} else if (d <= -1.26e-17) {
		tmp = t_3 * Math.sqrt((d / h));
	} else if (d <= 4.05e-299) {
		tmp = d / (Math.exp(Math.log1p(t_0)) + -1.0);
	} else {
		tmp = t_1 * (1.0 - ((0.5 * (h / l)) * Math.pow((M * ((0.5 * D) / d)), 2.0)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((l * h))
	t_1 = d / t_0
	t_2 = math.pow((d / h), 0.25)
	t_3 = math.sqrt((d / l))
	tmp = 0
	if d <= -6e+126:
		tmp = t_2 * (t_2 * t_3)
	elif d <= -1.85e+73:
		tmp = math.log1p(math.expm1(t_1))
	elif d <= -1.26e-17:
		tmp = t_3 * math.sqrt((d / h))
	elif d <= 4.05e-299:
		tmp = d / (math.exp(math.log1p(t_0)) + -1.0)
	else:
		tmp = t_1 * (1.0 - ((0.5 * (h / l)) * math.pow((M * ((0.5 * D) / d)), 2.0)))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(d / t_0)
	t_2 = Float64(d / h) ^ 0.25
	t_3 = sqrt(Float64(d / l))
	tmp = 0.0
	if (d <= -6e+126)
		tmp = Float64(t_2 * Float64(t_2 * t_3));
	elseif (d <= -1.85e+73)
		tmp = log1p(expm1(t_1));
	elseif (d <= -1.26e-17)
		tmp = Float64(t_3 * sqrt(Float64(d / h)));
	elseif (d <= 4.05e-299)
		tmp = Float64(d / Float64(exp(log1p(t_0)) + -1.0));
	else
		tmp = Float64(t_1 * Float64(1.0 - Float64(Float64(0.5 * Float64(h / l)) * (Float64(M * Float64(Float64(0.5 * D) / d)) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(d / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(d / h), $MachinePrecision], 0.25], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[d, -6e+126], N[(t$95$2 * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.85e+73], N[Log[1 + N[(Exp[t$95$1] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -1.26e-17], N[(t$95$3 * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.05e-299], N[(d / N[(N[Exp[N[Log[1 + t$95$0], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(1.0 - N[(N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(M * N[(N[(0.5 * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -6.0000000000000005e126

   1. Initial program 75.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 75.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 4.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 4.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 4.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 4.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 0.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 0.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 0.5%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 0.5%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 0.5%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 0.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 0.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 4.0%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 4.0%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 4.0%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 4.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. unpow1/2 0.0%

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. metadata-eval 0.0%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. sqrt-div 62.9%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. *-commutative 62.9%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

       9. add-sqr-sqrt 62.9%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\left(\sqrt{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \cdot \sqrt{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right)} \]

       10. associate-*r* 62.9%

           \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \sqrt{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}}} \]

       11. sqrt-pow1 63.0%

           \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}}\right) \cdot \sqrt{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

       12. metadata-eval 63.0%

           \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)}\right) \cdot \sqrt{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

       13. metadata-eval 63.0%

           \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.25}}\right) \cdot \sqrt{{\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

       14. sqrt-pow1 62.9%

           \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right) \cdot \color{blue}{{\left(\frac{d}{h}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}} \]

       15. metadata-eval 62.9%

           \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{\color{blue}{0.5}}{2}\right)} \]

       16. metadata-eval 62.9%

           \[\leadsto \left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right) \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.25}} \]

   12. Applied egg-rr 62.9%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{0.25}\right) \cdot {\left(\frac{d}{h}\right)}^{0.25}} \]

   if -6.0000000000000005e126 < d < -1.84999999999999987e73

   1. Initial program 80.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 9.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 9.6%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 9.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 9.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 0.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 0.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 0.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 0.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 9.6%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 9.6%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 9.6%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 9.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. log1p-expm1-u 54.1%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

   12. Applied egg-rr 54.1%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

   if -1.84999999999999987e73 < d < -1.2600000000000001e-17

   1. Initial program 81.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 81.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 3.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 3.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 3.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 3.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 2.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 3.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 3.0%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 3.0%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 3.0%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 3.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 2.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 3.2%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 3.2%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 3.2%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 3.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. unpow1/2 0.0%

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. metadata-eval 0.0%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. sqrt-div 64.5%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. *-commutative 64.5%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

       9. metadata-eval 64.5%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \]

       10. unpow1/2 64.5%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

   12. Applied egg-rr 64.5%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   if -1.2600000000000001e-17 < d < 4.0499999999999998e-299

   1. Initial program 58.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 58.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 19.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 19.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 19.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 19.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 6.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 6.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 6.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 6.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 6.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 19.2%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 19.2%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 19.2%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 19.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 19.2%

          \[\leadsto \frac{d}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{h \cdot \ell}\right)\right)}} \]

       2. expm1-udef 40.7%

          \[\leadsto \frac{d}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{h \cdot \ell}\right)} - 1}} \]

   12. Applied egg-rr 40.7%

       \[\leadsto \frac{d}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt{h \cdot \ell}\right)} - 1}} \]

   if 4.0499999999999998e-299 < d

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}} \cdot \sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}}\right) \]

      2. pow2 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\color{blue}{\left(\sqrt{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. *-commutative 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 66.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{1}{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. sqrt-prod 66.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{1}{2}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. sqrt-pow1 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. pow1 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. div-inv 69.8%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. metadata-eval 69.8%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. metadata-eval 69.8%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\color{blue}{0.5}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   3. Applied egg-rr 69.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   4. Step-by-step derivation

      1. expm1-log1p-u 68.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. expm1-udef 45.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. metadata-eval 45.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. unpow1/2 45.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{\ell}}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. Applied egg-rr 45.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]
   6. Step-by-step derivation

      1. expm1-def 68.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. expm1-log1p 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. Simplified 69.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. Applied egg-rr 62.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} + \left(\left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. *-lft-identity 62.8%

          \[\leadsto \color{blue}{1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} + \left(\left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

       2. distribute-rgt-in 71.6%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)\right)} \]

       3. *-commutative 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\left(-\frac{h}{\ell}\right) \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}\right) \]

       4. cancel-sign-sub-inv 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)} \]

       5. associate-*r* 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}\right) \]

       6. associate-*r/ 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(M \cdot \color{blue}{\frac{0.5 \cdot D}{d}}\right)}^{2}\right) \]

   11. Simplified 71.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{+126}:\\ \;\;\;\;{\left(\frac{d}{h}\right)}^{0.25} \cdot \left({\left(\frac{d}{h}\right)}^{0.25} \cdot \sqrt{\frac{d}{\ell}}\right)\\ \mathbf{elif}\;d \leq -1.85 \cdot 10^{+73}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)\\ \mathbf{elif}\;d \leq -1.26 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-299}:\\ \;\;\;\;\frac{d}{e^{\mathsf{log1p}\left(\sqrt{\ell \cdot h}\right)} + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)\\ \end{array} \]

Alternative 14: 53.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \sqrt{\ell \cdot h}\\ t_1 := \frac{d}{t_0}\\ t_2 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -1.66 \cdot 10^{+125}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -3.15 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-17}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-299}:\\ \;\;\;\;\frac{d}{\log \left(e^{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (sqrt (* l h)))
        (t_1 (/ d t_0))
        (t_2 (* (sqrt (/ d l)) (sqrt (/ d h)))))
   (if (<= d -1.66e+125)
     t_2
     (if (<= d -3.15e+72)
       (log1p (expm1 t_1))
       (if (<= d -1.3e-17)
         t_2
         (if (<= d 4.05e-299)
           (/ d (log (exp t_0)))
           (*
            t_1
            (- 1.0 (* (* 0.5 (/ h l)) (pow (* M (/ (* 0.5 D) d)) 2.0))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = sqrt((l * h));
	double t_1 = d / t_0;
	double t_2 = sqrt((d / l)) * sqrt((d / h));
	double tmp;
	if (d <= -1.66e+125) {
		tmp = t_2;
	} else if (d <= -3.15e+72) {
		tmp = log1p(expm1(t_1));
	} else if (d <= -1.3e-17) {
		tmp = t_2;
	} else if (d <= 4.05e-299) {
		tmp = d / log(exp(t_0));
	} else {
		tmp = t_1 * (1.0 - ((0.5 * (h / l)) * pow((M * ((0.5 * D) / d)), 2.0)));
	}
	return tmp;
}

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = Math.sqrt((l * h));
	double t_1 = d / t_0;
	double t_2 = Math.sqrt((d / l)) * Math.sqrt((d / h));
	double tmp;
	if (d <= -1.66e+125) {
		tmp = t_2;
	} else if (d <= -3.15e+72) {
		tmp = Math.log1p(Math.expm1(t_1));
	} else if (d <= -1.3e-17) {
		tmp = t_2;
	} else if (d <= 4.05e-299) {
		tmp = d / Math.log(Math.exp(t_0));
	} else {
		tmp = t_1 * (1.0 - ((0.5 * (h / l)) * Math.pow((M * ((0.5 * D) / d)), 2.0)));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	t_0 = math.sqrt((l * h))
	t_1 = d / t_0
	t_2 = math.sqrt((d / l)) * math.sqrt((d / h))
	tmp = 0
	if d <= -1.66e+125:
		tmp = t_2
	elif d <= -3.15e+72:
		tmp = math.log1p(math.expm1(t_1))
	elif d <= -1.3e-17:
		tmp = t_2
	elif d <= 4.05e-299:
		tmp = d / math.log(math.exp(t_0))
	else:
		tmp = t_1 * (1.0 - ((0.5 * (h / l)) * math.pow((M * ((0.5 * D) / d)), 2.0)))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = sqrt(Float64(l * h))
	t_1 = Float64(d / t_0)
	t_2 = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)))
	tmp = 0.0
	if (d <= -1.66e+125)
		tmp = t_2;
	elseif (d <= -3.15e+72)
		tmp = log1p(expm1(t_1));
	elseif (d <= -1.3e-17)
		tmp = t_2;
	elseif (d <= 4.05e-299)
		tmp = Float64(d / log(exp(t_0)));
	else
		tmp = Float64(t_1 * Float64(1.0 - Float64(Float64(0.5 * Float64(h / l)) * (Float64(M * Float64(Float64(0.5 * D) / d)) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(d / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.66e+125], t$95$2, If[LessEqual[d, -3.15e+72], N[Log[1 + N[(Exp[t$95$1] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -1.3e-17], t$95$2, If[LessEqual[d, 4.05e-299], N[(d / N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(1.0 - N[(N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(M * N[(N[(0.5 * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.66000000000000005e125 or -3.14999999999999981e72 < d < -1.30000000000000002e-17

   1. Initial program 77.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 3.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 3.8%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 3.8%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 3.8%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 1.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 1.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 1.2%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 1.2%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 1.2%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 1.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 1.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 3.8%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 3.8%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 3.8%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 3.8%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. unpow1/2 0.0%

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. metadata-eval 0.0%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. sqrt-div 63.4%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. *-commutative 63.4%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

       9. metadata-eval 63.4%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \]

       10. unpow1/2 63.4%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

   12. Applied egg-rr 63.4%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   if -1.66000000000000005e125 < d < -3.14999999999999981e72

   1. Initial program 80.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 9.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 9.6%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 9.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 9.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 0.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 0.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 0.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 0.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 9.6%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 9.6%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 9.6%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 9.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. log1p-expm1-u 54.1%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

   12. Applied egg-rr 54.1%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

   if -1.30000000000000002e-17 < d < 4.0499999999999998e-299

   1. Initial program 58.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 58.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 19.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 19.2%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 19.2%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 19.2%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 6.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 6.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 6.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 6.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 6.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 7.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 19.2%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 19.2%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 19.2%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 19.2%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-log-exp 40.7%

          \[\leadsto \frac{d}{\color{blue}{\log \left(e^{\sqrt{h \cdot \ell}}\right)}} \]

   12. Applied egg-rr 40.7%

       \[\leadsto \frac{d}{\color{blue}{\log \left(e^{\sqrt{h \cdot \ell}}\right)}} \]

   if 4.0499999999999998e-299 < d

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}} \cdot \sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}}\right) \]

      2. pow2 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\color{blue}{\left(\sqrt{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. *-commutative 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 66.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{1}{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. sqrt-prod 66.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{1}{2}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. sqrt-pow1 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. pow1 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. div-inv 69.8%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. metadata-eval 69.8%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. metadata-eval 69.8%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\color{blue}{0.5}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   3. Applied egg-rr 69.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   4. Step-by-step derivation

      1. expm1-log1p-u 68.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. expm1-udef 45.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. metadata-eval 45.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. unpow1/2 45.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{\ell}}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. Applied egg-rr 45.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]
   6. Step-by-step derivation

      1. expm1-def 68.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. expm1-log1p 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. Simplified 69.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. Applied egg-rr 62.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} + \left(\left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. *-lft-identity 62.8%

          \[\leadsto \color{blue}{1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} + \left(\left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

       2. distribute-rgt-in 71.6%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)\right)} \]

       3. *-commutative 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\left(-\frac{h}{\ell}\right) \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}\right) \]

       4. cancel-sign-sub-inv 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)} \]

       5. associate-*r* 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}\right) \]

       6. associate-*r/ 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(M \cdot \color{blue}{\frac{0.5 \cdot D}{d}}\right)}^{2}\right) \]

   11. Simplified 71.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.66 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -3.15 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-299}:\\ \;\;\;\;\frac{d}{\log \left(e^{\sqrt{\ell \cdot h}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)\\ \end{array} \]

Alternative 15: 53.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} t_0 := \frac{d}{\sqrt{\ell \cdot h}}\\ t_1 := \sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{if}\;d \leq -1 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-299}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (let* ((t_0 (/ d (sqrt (* l h)))) (t_1 (* (sqrt (/ d l)) (sqrt (/ d h)))))
   (if (<= d -1e+125)
     t_1
     (if (<= d -4.6e+67)
       (log1p (expm1 t_0))
       (if (<= d -6e-140)
         t_1
         (if (<= d 4.05e-299)
           (/ d (cbrt (pow (* l h) 1.5)))
           (*
            t_0
            (- 1.0 (* (* 0.5 (/ h l)) (pow (* M (/ (* 0.5 D) d)) 2.0))))))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double t_0 = d / sqrt((l * h));
	double t_1 = sqrt((d / l)) * sqrt((d / h));
	double tmp;
	if (d <= -1e+125) {
		tmp = t_1;
	} else if (d <= -4.6e+67) {
		tmp = log1p(expm1(t_0));
	} else if (d <= -6e-140) {
		tmp = t_1;
	} else if (d <= 4.05e-299) {
		tmp = d / cbrt(pow((l * h), 1.5));
	} else {
		tmp = t_0 * (1.0 - ((0.5 * (h / l)) * pow((M * ((0.5 * D) / d)), 2.0)));
	}
	return tmp;
}

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double t_0 = d / Math.sqrt((l * h));
	double t_1 = Math.sqrt((d / l)) * Math.sqrt((d / h));
	double tmp;
	if (d <= -1e+125) {
		tmp = t_1;
	} else if (d <= -4.6e+67) {
		tmp = Math.log1p(Math.expm1(t_0));
	} else if (d <= -6e-140) {
		tmp = t_1;
	} else if (d <= 4.05e-299) {
		tmp = d / Math.cbrt(Math.pow((l * h), 1.5));
	} else {
		tmp = t_0 * (1.0 - ((0.5 * (h / l)) * Math.pow((M * ((0.5 * D) / d)), 2.0)));
	}
	return tmp;
}

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	t_0 = Float64(d / sqrt(Float64(l * h)))
	t_1 = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)))
	tmp = 0.0
	if (d <= -1e+125)
		tmp = t_1;
	elseif (d <= -4.6e+67)
		tmp = log1p(expm1(t_0));
	elseif (d <= -6e-140)
		tmp = t_1;
	elseif (d <= 4.05e-299)
		tmp = Float64(d / cbrt((Float64(l * h) ^ 1.5)));
	else
		tmp = Float64(t_0 * Float64(1.0 - Float64(Float64(0.5 * Float64(h / l)) * (Float64(M * Float64(Float64(0.5 * D) / d)) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := Block[{t$95$0 = N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1e+125], t$95$1, If[LessEqual[d, -4.6e+67], N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, -6e-140], t$95$1, If[LessEqual[d, 4.05e-299], N[(d / N[Power[N[Power[N[(l * h), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(1.0 - N[(N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(M * N[(N[(0.5 * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -9.9999999999999992e124 or -4.5999999999999997e67 < d < -6.00000000000000037e-140

   1. Initial program 77.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.3%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 7.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 7.0%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 7.0%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 7.0%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 1.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 1.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 1.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 1.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 1.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 1.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 1.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 7.0%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 7.0%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 7.0%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 7.0%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. unpow1/2 0.0%

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. metadata-eval 0.0%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. sqrt-div 51.1%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. *-commutative 51.1%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

       9. metadata-eval 51.1%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \]

       10. unpow1/2 51.1%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

   12. Applied egg-rr 51.1%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   if -9.9999999999999992e124 < d < -4.5999999999999997e67

   1. Initial program 80.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 9.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 9.6%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 9.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 9.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 0.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 0.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 0.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 0.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 9.6%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 9.6%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 9.6%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 9.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. log1p-expm1-u 54.1%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

   12. Applied egg-rr 54.1%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{h \cdot \ell}}\right)\right)} \]

   if -6.00000000000000037e-140 < d < 4.0499999999999998e-299

   1. Initial program 43.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 43.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 23.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 23.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 23.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 23.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 11.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 10.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 10.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 10.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 10.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 10.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 11.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 23.4%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 23.4%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 23.4%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 23.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-cbrt-cube 38.2%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}}}} \]

       2. pow1/3 38.2%

          \[\leadsto \frac{d}{\color{blue}{{\left(\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}}} \]

       3. add-sqr-sqrt 38.2%

          \[\leadsto \frac{d}{{\left(\color{blue}{\left(h \cdot \ell\right)} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       4. pow1 38.2%

          \[\leadsto \frac{d}{{\left(\color{blue}{{\left(h \cdot \ell\right)}^{1}} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       5. pow1/2 38.2%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{1} \cdot \color{blue}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

       6. metadata-eval 38.2%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{1} \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right)}^{0.3333333333333333}} \]

       7. pow-prod-up 38.2%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(h \cdot \ell\right)}^{\left(1 + \frac{1}{2}\right)}\right)}}^{0.3333333333333333}} \]

       8. metadata-eval 38.2%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\left(1 + \color{blue}{0.5}\right)}\right)}^{0.3333333333333333}} \]

       9. metadata-eval 38.2%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   12. Applied egg-rr 38.2%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(h \cdot \ell\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   13. Step-by-step derivation

       1. unpow1/3 38.2%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   14. Simplified 38.2%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   if 4.0499999999999998e-299 < d

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}} \cdot \sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}}\right) \]

      2. pow2 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\color{blue}{\left(\sqrt{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. *-commutative 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 66.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{1}{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. sqrt-prod 66.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{1}{2}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. sqrt-pow1 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. pow1 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. div-inv 69.8%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. metadata-eval 69.8%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. metadata-eval 69.8%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\color{blue}{0.5}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   3. Applied egg-rr 69.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   4. Step-by-step derivation

      1. expm1-log1p-u 68.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. expm1-udef 45.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. metadata-eval 45.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. unpow1/2 45.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{\ell}}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. Applied egg-rr 45.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]
   6. Step-by-step derivation

      1. expm1-def 68.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. expm1-log1p 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. Simplified 69.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. Applied egg-rr 62.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} + \left(\left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. *-lft-identity 62.8%

          \[\leadsto \color{blue}{1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} + \left(\left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

       2. distribute-rgt-in 71.6%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)\right)} \]

       3. *-commutative 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\left(-\frac{h}{\ell}\right) \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}\right) \]

       4. cancel-sign-sub-inv 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)} \]

       5. associate-*r* 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}\right) \]

       6. associate-*r/ 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(M \cdot \color{blue}{\frac{0.5 \cdot D}{d}}\right)}^{2}\right) \]

   11. Simplified 71.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{+125}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq -4.6 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{d}{\sqrt{\ell \cdot h}}\right)\right)\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-299}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)\\ \end{array} \]

Alternative 16: 54.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-299}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.8e-139)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= d 4.05e-299)
     (/ d (cbrt (pow (* l h) 1.5)))
     (*
      (/ d (sqrt (* l h)))
      (- 1.0 (* (* 0.5 (/ h l)) (pow (* M (/ (* 0.5 D) d)) 2.0)))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.8e-139) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= 4.05e-299) {
		tmp = d / cbrt(pow((l * h), 1.5));
	} else {
		tmp = (d / sqrt((l * h))) * (1.0 - ((0.5 * (h / l)) * pow((M * ((0.5 * D) / d)), 2.0)));
	}
	return tmp;
}

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.8e-139) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (d <= 4.05e-299) {
		tmp = d / Math.cbrt(Math.pow((l * h), 1.5));
	} else {
		tmp = (d / Math.sqrt((l * h))) * (1.0 - ((0.5 * (h / l)) * Math.pow((M * ((0.5 * D) / d)), 2.0)));
	}
	return tmp;
}

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.8e-139)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= 4.05e-299)
		tmp = Float64(d / cbrt((Float64(l * h) ^ 1.5)));
	else
		tmp = Float64(Float64(d / sqrt(Float64(l * h))) * Float64(1.0 - Float64(Float64(0.5 * Float64(h / l)) * (Float64(M * Float64(Float64(0.5 * D) / d)) ^ 2.0))));
	end
	return tmp
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.8e-139], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.05e-299], N[(d / N[Power[N[Power[N[(l * h), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[(0.5 * N[(h / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(M * N[(N[(0.5 * D), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.7999999999999999e-139

   1. Initial program 77.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 7.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 7.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 7.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 7.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 1.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 1.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 1.1%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 1.1%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 1.1%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 1.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 1.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 7.4%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 7.4%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 7.4%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 7.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. unpow1/2 0.0%

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. metadata-eval 0.0%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. sqrt-div 46.3%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. *-commutative 46.3%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

       9. metadata-eval 46.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \]

       10. unpow1/2 46.3%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

   12. Applied egg-rr 46.3%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   if -2.7999999999999999e-139 < d < 4.0499999999999998e-299

   1. Initial program 43.5%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 43.5%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 23.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 23.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 23.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 23.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 11.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 10.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 10.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 10.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 10.7%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 10.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 11.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 23.4%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 23.4%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 23.4%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 23.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-cbrt-cube 38.2%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}}}} \]

       2. pow1/3 38.2%

          \[\leadsto \frac{d}{\color{blue}{{\left(\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}}} \]

       3. add-sqr-sqrt 38.2%

          \[\leadsto \frac{d}{{\left(\color{blue}{\left(h \cdot \ell\right)} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       4. pow1 38.2%

          \[\leadsto \frac{d}{{\left(\color{blue}{{\left(h \cdot \ell\right)}^{1}} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       5. pow1/2 38.2%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{1} \cdot \color{blue}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

       6. metadata-eval 38.2%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{1} \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right)}^{0.3333333333333333}} \]

       7. pow-prod-up 38.2%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(h \cdot \ell\right)}^{\left(1 + \frac{1}{2}\right)}\right)}}^{0.3333333333333333}} \]

       8. metadata-eval 38.2%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\left(1 + \color{blue}{0.5}\right)}\right)}^{0.3333333333333333}} \]

       9. metadata-eval 38.2%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   12. Applied egg-rr 38.2%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(h \cdot \ell\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   13. Step-by-step derivation

       1. unpow1/3 38.2%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   14. Simplified 38.2%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   if 4.0499999999999998e-299 < d

   1. Initial program 67.3%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}} \cdot \sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}}\right) \]

      2. pow2 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\sqrt{\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}}\right)}^{2}}\right) \]

      3. sqrt-prod 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\color{blue}{\left(\sqrt{\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}}^{2}\right) \]

      4. *-commutative 67.3%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      5. frac-times 66.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\sqrt{{\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{1}{2}} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      6. sqrt-prod 66.5%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\color{blue}{\left(\sqrt{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}} \cdot \sqrt{\frac{1}{2}}\right)} \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      7. sqrt-pow1 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      8. metadata-eval 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\color{blue}{1}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      9. pow1 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)} \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      10. div-inv 69.8%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\color{blue}{\left(M \cdot \frac{1}{2}\right)} \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      11. metadata-eval 69.8%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot \color{blue}{0.5}\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      12. metadata-eval 69.8%

          \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{\color{blue}{0.5}}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   3. Applied egg-rr 69.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{{\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}}\right) \]
   4. Step-by-step derivation

      1. expm1-log1p-u 68.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. expm1-udef 45.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      3. metadata-eval 45.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left({\left(\frac{d}{\ell}\right)}^{\color{blue}{0.5}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      4. unpow1/2 45.4%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{d}{\ell}}}\right)} - 1\right)\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   5. Applied egg-rr 45.4%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)} - 1\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]
   6. Step-by-step derivation

      1. expm1-def 68.9%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{d}{\ell}}\right)\right)}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

      2. expm1-log1p 69.8%

         \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   7. Simplified 69.8%

      \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}}\right) \cdot \left(1 - {\left(\left(\left(\left(M \cdot 0.5\right) \cdot \frac{D}{d}\right) \cdot \sqrt{0.5}\right) \cdot \sqrt{\frac{h}{\ell}}\right)}^{2}\right) \]

   9. Applied egg-rr 62.8%

      \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} + \left(\left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}}} \]
   10. Step-by-step derivation

       1. *-lft-identity 62.8%

          \[\leadsto \color{blue}{1 \cdot \frac{d}{\sqrt{h \cdot \ell}}} + \left(\left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)\right) \cdot \frac{d}{\sqrt{h \cdot \ell}} \]

       2. distribute-rgt-in 71.6%

          \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right) \cdot \left(-\frac{h}{\ell}\right)\right)} \]

       3. *-commutative 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 + \color{blue}{\left(-\frac{h}{\ell}\right) \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)}\right) \]

       4. cancel-sign-sub-inv 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \color{blue}{\left(1 - \frac{h}{\ell} \cdot \left(0.5 \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}\right)\right)} \]

       5. associate-*r* 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \color{blue}{\left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(M \cdot \left(0.5 \cdot \frac{D}{d}\right)\right)}^{2}}\right) \]

       6. associate-*r/ 71.6%

          \[\leadsto \frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(M \cdot \color{blue}{\frac{0.5 \cdot D}{d}}\right)}^{2}\right) \]

   11. Simplified 71.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}} \cdot \left(1 - \left(\frac{h}{\ell} \cdot 0.5\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 4.05 \cdot 10^{-299}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\sqrt{\ell \cdot h}} \cdot \left(1 - \left(0.5 \cdot \frac{h}{\ell}\right) \cdot {\left(M \cdot \frac{0.5 \cdot D}{d}\right)}^{2}\right)\\ \end{array} \]

Alternative 17: 38.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -1.42 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-66}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -1.42e-142)
   (sqrt (/ (pow d 2.0) (* l h)))
   (if (<= d 3.8e-66)
     (* d (sqrt (/ (/ 1.0 l) h)))
     (* d (/ (pow l -0.5) (sqrt h))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.42e-142) {
		tmp = sqrt((pow(d, 2.0) / (l * h)));
	} else if (d <= 3.8e-66) {
		tmp = d * sqrt(((1.0 / l) / h));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-1.42d-142)) then
        tmp = sqrt(((d ** 2.0d0) / (l * h)))
    else if (d <= 3.8d-66) then
        tmp = d * sqrt(((1.0d0 / l) / h))
    else
        tmp = d * ((l ** (-0.5d0)) / sqrt(h))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -1.42e-142) {
		tmp = Math.sqrt((Math.pow(d, 2.0) / (l * h)));
	} else if (d <= 3.8e-66) {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -1.42e-142:
		tmp = math.sqrt((math.pow(d, 2.0) / (l * h)))
	elif d <= 3.8e-66:
		tmp = d * math.sqrt(((1.0 / l) / h))
	else:
		tmp = d * (math.pow(l, -0.5) / math.sqrt(h))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -1.42e-142)
		tmp = sqrt(Float64((d ^ 2.0) / Float64(l * h)));
	elseif (d <= 3.8e-66)
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -1.42e-142)
		tmp = sqrt(((d ^ 2.0) / (l * h)));
	elseif (d <= 3.8e-66)
		tmp = d * sqrt(((1.0 / l) / h));
	else
		tmp = d * ((l ^ -0.5) / sqrt(h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -1.42e-142], N[Sqrt[N[(N[Power[d, 2.0], $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, 3.8e-66], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.42000000000000007e-142

   1. Initial program 77.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 7.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 7.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 7.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 7.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 1.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 1.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 1.1%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 1.1%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 1.1%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 1.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 1.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 7.4%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 7.4%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 7.4%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 7.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. unpow1/2 0.0%

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. metadata-eval 0.0%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. sqrt-div 46.3%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. pow1/2 46.3%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}} \]

       9. metadata-eval 46.3%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \]

       10. pow-prod-down 35.4%

           \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   12. Applied egg-rr 29.1%

       \[\leadsto \color{blue}{{\left(\frac{{d}^{2}}{h \cdot \ell}\right)}^{0.5}} \]
   13. Step-by-step derivation

       1. unpow1/2 29.1%

          \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   14. Simplified 29.1%

       \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   if -1.42000000000000007e-142 < d < 3.7999999999999998e-66

   1. Initial program 48.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 47.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 22.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 22.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 23.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 23.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]

   if 3.7999999999999998e-66 < d

   1. Initial program 78.6%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 78.6%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 52.5%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 52.5%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 53.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 53.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. sqrt-div 58.9%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]

      2. inv-pow 58.9%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{\ell}^{-1}}}}{\sqrt{h}} \]

      3. sqrt-pow1 58.9%

         \[\leadsto d \cdot \frac{\color{blue}{{\ell}^{\left(\frac{-1}{2}\right)}}}{\sqrt{h}} \]

      4. metadata-eval 58.9%

         \[\leadsto d \cdot \frac{{\ell}^{\color{blue}{-0.5}}}{\sqrt{h}} \]

   8. Applied egg-rr 58.9%

      \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.42 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-66}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 18: 39.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-266}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -9e-140)
   (sqrt (/ (pow d 2.0) (* l h)))
   (if (<= d 7.8e-266)
     (/ d (cbrt (pow (* l h) 1.5)))
     (* d (/ (pow l -0.5) (sqrt h))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -9e-140) {
		tmp = sqrt((pow(d, 2.0) / (l * h)));
	} else if (d <= 7.8e-266) {
		tmp = d / cbrt(pow((l * h), 1.5));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -9e-140) {
		tmp = Math.sqrt((Math.pow(d, 2.0) / (l * h)));
	} else if (d <= 7.8e-266) {
		tmp = d / Math.cbrt(Math.pow((l * h), 1.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -9e-140)
		tmp = sqrt(Float64((d ^ 2.0) / Float64(l * h)));
	elseif (d <= 7.8e-266)
		tmp = Float64(d / cbrt((Float64(l * h) ^ 1.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -9e-140], N[Sqrt[N[(N[Power[d, 2.0], $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[d, 7.8e-266], N[(d / N[Power[N[Power[N[(l * h), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.00000000000000008e-140

   1. Initial program 77.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 7.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 7.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 7.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 7.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 1.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 1.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 1.1%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 1.1%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 1.1%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 1.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 1.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 7.4%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 7.4%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 7.4%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 7.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. unpow1/2 0.0%

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. metadata-eval 0.0%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. sqrt-div 46.3%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. pow1/2 46.3%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}} \]

       9. metadata-eval 46.3%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \]

       10. pow-prod-down 35.4%

           \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   12. Applied egg-rr 29.1%

       \[\leadsto \color{blue}{{\left(\frac{{d}^{2}}{h \cdot \ell}\right)}^{0.5}} \]
   13. Step-by-step derivation

       1. unpow1/2 29.1%

          \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   14. Simplified 29.1%

       \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   if -9.00000000000000008e-140 < d < 7.80000000000000057e-266

   1. Initial program 44.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 44.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 23.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 23.6%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 23.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 23.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 12.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 10.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 10.0%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 10.0%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 10.0%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 10.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 12.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 23.6%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 23.6%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 23.6%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 23.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-cbrt-cube 36.9%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}}}} \]

       2. pow1/3 36.7%

          \[\leadsto \frac{d}{\color{blue}{{\left(\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}}} \]

       3. add-sqr-sqrt 36.7%

          \[\leadsto \frac{d}{{\left(\color{blue}{\left(h \cdot \ell\right)} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       4. pow1 36.7%

          \[\leadsto \frac{d}{{\left(\color{blue}{{\left(h \cdot \ell\right)}^{1}} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       5. pow1/2 36.7%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{1} \cdot \color{blue}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

       6. metadata-eval 36.7%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{1} \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right)}^{0.3333333333333333}} \]

       7. pow-prod-up 36.7%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(h \cdot \ell\right)}^{\left(1 + \frac{1}{2}\right)}\right)}}^{0.3333333333333333}} \]

       8. metadata-eval 36.7%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\left(1 + \color{blue}{0.5}\right)}\right)}^{0.3333333333333333}} \]

       9. metadata-eval 36.7%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   12. Applied egg-rr 36.7%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(h \cdot \ell\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   13. Step-by-step derivation

       1. unpow1/3 36.9%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   14. Simplified 36.9%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   if 7.80000000000000057e-266 < d

   1. Initial program 67.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 67.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 39.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 39.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 41.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 41.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. sqrt-div 44.4%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]

      2. inv-pow 44.4%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{\ell}^{-1}}}}{\sqrt{h}} \]

      3. sqrt-pow1 44.4%

         \[\leadsto d \cdot \frac{\color{blue}{{\ell}^{\left(\frac{-1}{2}\right)}}}{\sqrt{h}} \]

      4. metadata-eval 44.4%

         \[\leadsto d \cdot \frac{{\ell}^{\color{blue}{-0.5}}}{\sqrt{h}} \]

   8. Applied egg-rr 44.4%

      \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{\ell \cdot h}}\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-266}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 19: 45.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.45 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-266}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.45e-139)
   (* (sqrt (/ d l)) (sqrt (/ d h)))
   (if (<= d 7.8e-266)
     (/ d (cbrt (pow (* l h) 1.5)))
     (* d (/ (pow l -0.5) (sqrt h))))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.45e-139) {
		tmp = sqrt((d / l)) * sqrt((d / h));
	} else if (d <= 7.8e-266) {
		tmp = d / cbrt(pow((l * h), 1.5));
	} else {
		tmp = d * (pow(l, -0.5) / sqrt(h));
	}
	return tmp;
}

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.45e-139) {
		tmp = Math.sqrt((d / l)) * Math.sqrt((d / h));
	} else if (d <= 7.8e-266) {
		tmp = d / Math.cbrt(Math.pow((l * h), 1.5));
	} else {
		tmp = d * (Math.pow(l, -0.5) / Math.sqrt(h));
	}
	return tmp;
}

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.45e-139)
		tmp = Float64(sqrt(Float64(d / l)) * sqrt(Float64(d / h)));
	elseif (d <= 7.8e-266)
		tmp = Float64(d / cbrt((Float64(l * h) ^ 1.5)));
	else
		tmp = Float64(d * Float64((l ^ -0.5) / sqrt(h)));
	end
	return tmp
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.45e-139], N[(N[Sqrt[N[(d / l), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(d / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.8e-266], N[(d / N[Power[N[Power[N[(l * h), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(d * N[(N[Power[l, -0.5], $MachinePrecision] / N[Sqrt[h], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.45000000000000016e-139

   1. Initial program 77.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 7.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 7.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 7.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 7.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 1.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 1.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 1.1%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 1.1%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 1.1%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 1.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 1.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 7.4%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 7.4%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 7.4%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 7.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. unpow1/2 0.0%

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. metadata-eval 0.0%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. sqrt-div 46.3%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. *-commutative 46.3%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}} \]

       9. metadata-eval 46.3%

          \[\leadsto \sqrt{\frac{d}{\ell}} \cdot {\left(\frac{d}{h}\right)}^{\color{blue}{0.5}} \]

       10. unpow1/2 46.3%

           \[\leadsto \sqrt{\frac{d}{\ell}} \cdot \color{blue}{\sqrt{\frac{d}{h}}} \]

   12. Applied egg-rr 46.3%

       \[\leadsto \color{blue}{\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}} \]

   if -2.45000000000000016e-139 < d < 7.80000000000000057e-266

   1. Initial program 44.2%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 44.2%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 23.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 23.6%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 23.6%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 23.6%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 12.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 10.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 10.0%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 10.0%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 10.0%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 10.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 12.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 23.6%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 23.6%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 23.6%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 23.6%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-cbrt-cube 36.9%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}}}} \]

       2. pow1/3 36.7%

          \[\leadsto \frac{d}{\color{blue}{{\left(\left(\sqrt{h \cdot \ell} \cdot \sqrt{h \cdot \ell}\right) \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}}} \]

       3. add-sqr-sqrt 36.7%

          \[\leadsto \frac{d}{{\left(\color{blue}{\left(h \cdot \ell\right)} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       4. pow1 36.7%

          \[\leadsto \frac{d}{{\left(\color{blue}{{\left(h \cdot \ell\right)}^{1}} \cdot \sqrt{h \cdot \ell}\right)}^{0.3333333333333333}} \]

       5. pow1/2 36.7%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{1} \cdot \color{blue}{{\left(h \cdot \ell\right)}^{0.5}}\right)}^{0.3333333333333333}} \]

       6. metadata-eval 36.7%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{1} \cdot {\left(h \cdot \ell\right)}^{\color{blue}{\left(\frac{1}{2}\right)}}\right)}^{0.3333333333333333}} \]

       7. pow-prod-up 36.7%

          \[\leadsto \frac{d}{{\color{blue}{\left({\left(h \cdot \ell\right)}^{\left(1 + \frac{1}{2}\right)}\right)}}^{0.3333333333333333}} \]

       8. metadata-eval 36.7%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\left(1 + \color{blue}{0.5}\right)}\right)}^{0.3333333333333333}} \]

       9. metadata-eval 36.7%

          \[\leadsto \frac{d}{{\left({\left(h \cdot \ell\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

   12. Applied egg-rr 36.7%

       \[\leadsto \frac{d}{\color{blue}{{\left({\left(h \cdot \ell\right)}^{1.5}\right)}^{0.3333333333333333}}} \]
   13. Step-by-step derivation

       1. unpow1/3 36.9%

          \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   14. Simplified 36.9%

       \[\leadsto \frac{d}{\color{blue}{\sqrt[3]{{\left(h \cdot \ell\right)}^{1.5}}}} \]

   if 7.80000000000000057e-266 < d

   1. Initial program 67.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 67.0%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 39.9%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 39.9%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 41.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 41.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. sqrt-div 44.4%

         \[\leadsto d \cdot \color{blue}{\frac{\sqrt{\frac{1}{\ell}}}{\sqrt{h}}} \]

      2. inv-pow 44.4%

         \[\leadsto d \cdot \frac{\sqrt{\color{blue}{{\ell}^{-1}}}}{\sqrt{h}} \]

      3. sqrt-pow1 44.4%

         \[\leadsto d \cdot \frac{\color{blue}{{\ell}^{\left(\frac{-1}{2}\right)}}}{\sqrt{h}} \]

      4. metadata-eval 44.4%

         \[\leadsto d \cdot \frac{{\ell}^{\color{blue}{-0.5}}}{\sqrt{h}} \]

   8. Applied egg-rr 44.4%

      \[\leadsto d \cdot \color{blue}{\frac{{\ell}^{-0.5}}{\sqrt{h}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.45 \cdot 10^{-139}:\\ \;\;\;\;\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{-266}:\\ \;\;\;\;\frac{d}{\sqrt[3]{{\left(\ell \cdot h\right)}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \frac{{\ell}^{-0.5}}{\sqrt{h}}\\ \end{array} \]

Alternative 20: 35.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \begin{array}{l} \mathbf{if}\;d \leq -2.85 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D)
 :precision binary64
 (if (<= d -2.85e-140)
   (sqrt (/ (pow d 2.0) (* l h)))
   (* d (sqrt (/ (/ 1.0 l) h)))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.85e-140) {
		tmp = sqrt((pow(d, 2.0) / (l * h)));
	} else {
		tmp = d * sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    real(8) :: tmp
    if (d <= (-2.85d-140)) then
        tmp = sqrt(((d ** 2.0d0) / (l * h)))
    else
        tmp = d * sqrt(((1.0d0 / l) / h))
    end if
    code = tmp
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	double tmp;
	if (d <= -2.85e-140) {
		tmp = Math.sqrt((Math.pow(d, 2.0) / (l * h)));
	} else {
		tmp = d * Math.sqrt(((1.0 / l) / h));
	}
	return tmp;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	tmp = 0
	if d <= -2.85e-140:
		tmp = math.sqrt((math.pow(d, 2.0) / (l * h)))
	else:
		tmp = d * math.sqrt(((1.0 / l) / h))
	return tmp

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	tmp = 0.0
	if (d <= -2.85e-140)
		tmp = sqrt(Float64((d ^ 2.0) / Float64(l * h)));
	else
		tmp = Float64(d * sqrt(Float64(Float64(1.0 / l) / h)));
	end
	return tmp
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp_2 = code(d, h, l, M, D)
	tmp = 0.0;
	if (d <= -2.85e-140)
		tmp = sqrt(((d ^ 2.0) / (l * h)));
	else
		tmp = d * sqrt(((1.0 / l) / h));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := If[LessEqual[d, -2.85e-140], N[Sqrt[N[(N[Power[d, 2.0], $MachinePrecision] / N[(l * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.85e-140

   1. Initial program 77.8%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 77.8%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 7.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 7.4%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 7.4%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 7.4%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u 1.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

      2. expm1-udef 1.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

      3. associate-/l/ 1.1%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

      4. sqrt-div 1.1%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

      5. metadata-eval 1.1%

         \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

   8. Applied egg-rr 1.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 1.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

      2. expm1-log1p 7.4%

         \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

      3. associate-*r/ 7.4%

         \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

      4. *-rgt-identity 7.4%

         \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

   10. Simplified 7.4%

       \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 0.0%

          \[\leadsto \frac{\color{blue}{\sqrt{d} \cdot \sqrt{d}}}{\sqrt{h \cdot \ell}} \]

       2. sqrt-prod 0.0%

          \[\leadsto \frac{\sqrt{d} \cdot \sqrt{d}}{\color{blue}{\sqrt{h} \cdot \sqrt{\ell}}} \]

       3. times-frac 0.0%

          \[\leadsto \color{blue}{\frac{\sqrt{d}}{\sqrt{h}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}}} \]

       4. sqrt-div 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{d}{h}}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       5. unpow1/2 0.0%

          \[\leadsto \color{blue}{{\left(\frac{d}{h}\right)}^{0.5}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       6. metadata-eval 0.0%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \cdot \frac{\sqrt{d}}{\sqrt{\ell}} \]

       7. sqrt-div 46.3%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{\sqrt{\frac{d}{\ell}}} \]

       8. pow1/2 46.3%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{\left(\frac{d}{\ell}\right)}^{0.5}} \]

       9. metadata-eval 46.3%

          \[\leadsto {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\color{blue}{\left(\frac{1}{2}\right)}} \]

       10. pow-prod-down 35.4%

           \[\leadsto \color{blue}{{\left(\frac{d}{h} \cdot \frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \]

   12. Applied egg-rr 29.1%

       \[\leadsto \color{blue}{{\left(\frac{{d}^{2}}{h \cdot \ell}\right)}^{0.5}} \]
   13. Step-by-step derivation

       1. unpow1/2 29.1%

          \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   14. Simplified 29.1%

       \[\leadsto \color{blue}{\sqrt{\frac{{d}^{2}}{h \cdot \ell}}} \]

   if -2.85e-140 < d

   1. Initial program 62.4%

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

   3. Simplified 61.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

   4. Taylor expanded in d around inf 36.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
   5. Step-by-step derivation

      1. *-commutative 36.1%

         \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

      2. associate-/r* 37.1%

         \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

   6. Simplified 37.1%

      \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.85 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{\frac{{d}^{2}}{\ell \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\\ \end{array} \]

Alternative 21: 26.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* d (sqrt (/ (/ 1.0 l) h))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d * sqrt(((1.0 / l) / h));
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d * sqrt(((1.0d0 / l) / h))
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d * Math.sqrt(((1.0 / l) / h));
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * math.sqrt(((1.0 / l) / h))

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * sqrt(Float64(Float64(1.0 / l) / h)))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * sqrt(((1.0 / l) / h));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d * N[Sqrt[N[(N[(1.0 / l), $MachinePrecision] / h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.1%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 67.8%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

4. Taylor expanded in d around inf 25.4%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
5. Step-by-step derivation

   1. *-commutative 25.4%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   2. associate-/r* 26.0%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

6. Simplified 26.0%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]

7. Final simplification 26.0%

   \[\leadsto d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}} \]

Alternative 22: 26.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ \frac{d}{\sqrt{\ell \cdot h}} \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (/ d (sqrt (* l h))))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d / sqrt((l * h));
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d / sqrt((l * h))
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d / Math.sqrt((l * h));
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d / math.sqrt((l * h))

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d / sqrt(Float64(l * h)))
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d / sqrt((l * h));
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d / N[Sqrt[N[(l * h), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.1%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 67.8%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

4. Taylor expanded in d around inf 25.4%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
5. Step-by-step derivation

   1. *-commutative 25.4%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   2. associate-/r* 26.0%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

6. Simplified 26.0%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 21.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

   2. expm1-udef 15.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1} \]

   3. associate-/l/ 14.8%

      \[\leadsto e^{\mathsf{log1p}\left(d \cdot \sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1 \]

   4. sqrt-div 14.8%

      \[\leadsto e^{\mathsf{log1p}\left(d \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1 \]

   5. metadata-eval 14.8%

      \[\leadsto e^{\mathsf{log1p}\left(d \cdot \frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1 \]

8. Applied egg-rr 14.8%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 20.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(d \cdot \frac{1}{\sqrt{h \cdot \ell}}\right)\right)} \]

   2. expm1-log1p 25.3%

      \[\leadsto \color{blue}{d \cdot \frac{1}{\sqrt{h \cdot \ell}}} \]

   3. associate-*r/ 25.4%

      \[\leadsto \color{blue}{\frac{d \cdot 1}{\sqrt{h \cdot \ell}}} \]

   4. *-rgt-identity 25.4%

      \[\leadsto \frac{\color{blue}{d}}{\sqrt{h \cdot \ell}} \]

10. Simplified 25.4%

    \[\leadsto \color{blue}{\frac{d}{\sqrt{h \cdot \ell}}} \]

11. Final simplification 25.4%

    \[\leadsto \frac{d}{\sqrt{\ell \cdot h}} \]

Alternative 23: 4.6% accurate, 110.7× speedup

Math

\[\begin{array}{l} [M, D] = \mathsf{sort}([M, D])\\ \\ d \cdot 0 \end{array} \]

FPCore

NOTE: M and D should be sorted in increasing order before calling this function.
(FPCore (d h l M D) :precision binary64 (* d 0.0))

C

assert(M < D);
double code(double d, double h, double l, double M, double D) {
	return d * 0.0;
}

Fortran

NOTE: M and D should be sorted in increasing order before calling this function.
real(8) function code(d, h, l, m, d_1)
    real(8), intent (in) :: d
    real(8), intent (in) :: h
    real(8), intent (in) :: l
    real(8), intent (in) :: m
    real(8), intent (in) :: d_1
    code = d * 0.0d0
end function

Java

assert M < D;
public static double code(double d, double h, double l, double M, double D) {
	return d * 0.0;
}

Python

[M, D] = sort([M, D])
def code(d, h, l, M, D):
	return d * 0.0

Julia

M, D = sort([M, D])
function code(d, h, l, M, D)
	return Float64(d * 0.0)
end

MATLAB

M, D = num2cell(sort([M, D])){:}
function tmp = code(d, h, l, M, D)
	tmp = d * 0.0;
end

Wolfram

NOTE: M and D should be sorted in increasing order before calling this function.
code[d_, h_, l_, M_, D_] := N[(d * 0.0), $MachinePrecision]

Derivation

1. Initial program 68.1%

   \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right) \]

3. Simplified 67.8%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \mathsf{fma}\left({\left(\frac{-0.5 \cdot M}{d} \cdot D\right)}^{2}, -0.5 \cdot \frac{h}{\ell}, 1\right)} \]

4. Taylor expanded in d around inf 25.4%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{1}{h \cdot \ell}}} \]
5. Step-by-step derivation

   1. *-commutative 25.4%

      \[\leadsto d \cdot \sqrt{\frac{1}{\color{blue}{\ell \cdot h}}} \]

   2. associate-/r* 26.0%

      \[\leadsto d \cdot \sqrt{\color{blue}{\frac{\frac{1}{\ell}}{h}}} \]

6. Simplified 26.0%

   \[\leadsto \color{blue}{d \cdot \sqrt{\frac{\frac{1}{\ell}}{h}}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 25.6%

      \[\leadsto d \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)\right)} \]

   2. expm1-udef 18.1%

      \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{\frac{1}{\ell}}{h}}\right)} - 1\right)} \]

   3. associate-/l/ 18.1%

      \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1}{h \cdot \ell}}}\right)} - 1\right) \]

   4. sqrt-div 18.1%

      \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{1}}{\sqrt{h \cdot \ell}}}\right)} - 1\right) \]

   5. metadata-eval 18.1%

      \[\leadsto d \cdot \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{\sqrt{h \cdot \ell}}\right)} - 1\right) \]

8. Applied egg-rr 18.1%

   \[\leadsto d \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{h \cdot \ell}}\right)} - 1\right)} \]

9. Taylor expanded in h around inf 3.5%

   \[\leadsto d \cdot \left(\color{blue}{1} - 1\right) \]

10. Final simplification 3.5%

    \[\leadsto d \cdot 0 \]

Reproduce

herbie shell --seed 2023312 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))